126 results on '"Wilson-Cowan model"'
Search Results
2. A fractional-order Wilson-Cowan formulation of cortical disinhibition.
- Author
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González-Ramírez, L. R.
- Abstract
This work presents a fractional-order Wilson-Cowan model derivation under Caputo's formalism, considering an order of 0 < α ≤ 1 . To that end, we propose memory-dependent response functions and average neuronal excitation functions that permit us to naturally arrive at a fractional-order model that incorporates past dynamics into the description of synaptically coupled neuronal populations' activity. We then shift our focus on a particular example, aiming to analyze the fractional-order dynamics of the disinhibited cortex. This system mimics cortical activity observed during neurological disorders such as epileptic seizures, where an imbalance between excitation and inhibition is present, which allows brain dynamics to transition to a hyperexcited activity state. In the context of the first-order mathematical model, we recover traditional results showing a transition from a low-level activity state to a potentially pathological high-level activity state as an external factor modifies cortical inhibition. On the other hand, under the fractional-order formulation, we establish novel results showing that the system resists such transition as the order is decreased, permitting the possibility of staying in the low-activity state even with increased disinhibition. Furthermore, considering the memory index interpretation of the fractional-order model motivation here developed, our results establish that by increasing the memory index, the system becomes more resistant to transitioning towards the high-level activity state. That is, one possible effect of the memory index is to stabilize neuronal activity. Noticeably, this neuronal stabilizing effect is similar to homeostatic plasticity mechanisms. To summarize our results, we present a two-parameter structural portrait describing the system's dynamics dependent on a proposed disinhibition parameter and the order. We also explore numerical model simulations to validate our results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
3. Metabolic energetics underlying attractors in neural models.
- Author
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Buxton, Richard B. and Wong, Eric C.
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LIMIT cycles , *FLEXIBLE printed circuits , *CIRCUIT elements , *ACTION potentials , *ENERGY industries , *NEURAL codes - Abstract
Neural population modeling, including the role of neural attractors, is a promising tool for understanding many aspects of brain function. We propose a modeling framework to connect the abstract variables used in modeling to recent cellular-level estimates of the bioenergetic costs of different aspects of neural activity, measured in ATP consumed per second per neuron. Based on recent work, an empirical reference for brain ATP use for the awake resting brain was estimated as ~2 x 109 ATP/s-neuron across several mammalian species. The energetics framework was applied to the Wilson-Cowan (WC) model of two interacting populations of neurons, one excitatory (E) and one inhibitory (I). Attractors were considered to exhibit steady-state behavior and limit cycle behavior, both of which end when the excitatory stimulus ends, and sustained activity that persists after the stimulus ends. The energy cost of limit cycles, with oscillations much faster than the average neuronal firing rate of the population, is tracked more closely with the firing rate than the limit cycle frequency. Self-sustained firing driven by recurrent excitation, though, involves higher firing rates and a higher energy cost. As an example of a simple network in which each node is a WC model, a combination of three nodes can serve as a flexible circuit element that turns on with an oscillating output when input passes a threshold and then persists after the input ends (an "on-switch"), with moderate overall ATP use. The proposed framework can serve as a guide for anchoring neural population models to plausible bioenergetics requirements. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
4. Sub-harmonic entrainment of cortical gamma oscillations to deep brain stimulation in Parkinson's disease: Model based predictions and validation in three human subjects
- Author
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James J. Sermon, Maria Olaru, Juan Ansó, Stephanie Cernera, Simon Little, Maria Shcherbakova, Rafal Bogacz, Philip A. Starr, Timothy Denison, and Benoit Duchet
- Subjects
Sub-harmonic entrainment ,Deep brain stimulation ,Cortical gamma oscillations ,Parkinson's disease ,Wilson-Cowan model ,Neurosciences. Biological psychiatry. Neuropsychiatry ,RC321-571 - Abstract
Objectives: The exact mechanisms of deep brain stimulation (DBS) are still an active area of investigation, in spite of its clinical successes. This is due in part to the lack of understanding of the effects of stimulation on neuronal rhythms. Entrainment of brain oscillations has been hypothesised as a potential mechanism of neuromodulation. A better understanding of entrainment might further inform existing methods of continuous DBS, and help refine algorithms for adaptive methods. The purpose of this study is to develop and test a theoretical framework to predict entrainment of cortical rhythms to DBS across a wide range of stimulation parameters. Materials and Methods: We fit a model of interacting neural populations to selected features characterising PD patients' off-stimulation finely-tuned gamma rhythm recorded through electrocorticography. Using the fitted models, we predict basal ganglia DBS parameters that would result in 1:2 entrainment, a special case of sub-harmonic entrainment observed in patients and predicted by theory. Results: We show that the neural circuit models fitted to patient data exhibit 1:2 entrainment when stimulation is provided across a range of stimulation parameters. Furthermore, we verify key features of the region of 1:2 entrainment in the stimulation frequency/amplitude space with follow-up recordings from the same patients, such as the loss of 1:2 entrainment above certain stimulation amplitudes. Conclusion: Our results reveal that continuous, constant frequency DBS in patients may lead to nonlinear patterns of neuronal entrainment across stimulation parameters, and that these responses can be predicted by modelling. Should entrainment prove to be an important mechanism of therapeutic stimulation, our modelling framework may reduce the parameter space that clinicians must consider when programming devices for optimal benefit.
- Published
- 2023
- Full Text
- View/download PDF
5. Sub-harmonic entrainment of cortical gamma oscillations to deep brain stimulation in Parkinson's disease: Model based predictions and validation in three human subjects.
- Author
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Sermon, James J., Olaru, Maria, Ansó, Juan, Cernera, Stephanie, Little, Simon, Shcherbakova, Maria, Bogacz, Rafal, Starr, Philip A., Denison, Timothy, and Duchet, Benoit
- Abstract
The exact mechanisms of deep brain stimulation (DBS) are still an active area of investigation, in spite of its clinical successes. This is due in part to the lack of understanding of the effects of stimulation on neuronal rhythms. Entrainment of brain oscillations has been hypothesised as a potential mechanism of neuromodulation. A better understanding of entrainment might further inform existing methods of continuous DBS, and help refine algorithms for adaptive methods. The purpose of this study is to develop and test a theoretical framework to predict entrainment of cortical rhythms to DBS across a wide range of stimulation parameters. We fit a model of interacting neural populations to selected features characterising PD patients' off-stimulation finely-tuned gamma rhythm recorded through electrocorticography. Using the fitted models, we predict basal ganglia DBS parameters that would result in 1:2 entrainment, a special case of sub-harmonic entrainment observed in patients and predicted by theory. We show that the neural circuit models fitted to patient data exhibit 1:2 entrainment when stimulation is provided across a range of stimulation parameters. Furthermore, we verify key features of the region of 1:2 entrainment in the stimulation frequency/amplitude space with follow-up recordings from the same patients, such as the loss of 1:2 entrainment above certain stimulation amplitudes. Our results reveal that continuous, constant frequency DBS in patients may lead to nonlinear patterns of neuronal entrainment across stimulation parameters, and that these responses can be predicted by modelling. Should entrainment prove to be an important mechanism of therapeutic stimulation, our modelling framework may reduce the parameter space that clinicians must consider when programming devices for optimal benefit. • Entrainment of brain rhythms at half stimulation frequency is predicted by theory. • Models fitted to off-stimulation patient data predict 1:2 synchronisation regions. • Observations of 1:2 entrainment in patients are in line with model predictions. • 1:2 entrainment is biased to low stimulation frequencies and lost at high amplitudes. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
6. Hierarchical Wilson–Cowan Models and Connection Matrices.
- Author
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Zúñiga-Galindo, W. A. and Zambrano-Luna, B. A.
- Subjects
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ABELIAN groups , *MATRICES (Mathematics) , *CAUCHY problem , *COMPACT groups - Abstract
This work aims to study the interplay between the Wilson–Cowan model and connection matrices. These matrices describe cortical neural wiring, while Wilson–Cowan equations provide a dynamical description of neural interaction. We formulate Wilson–Cowan equations on locally compact Abelian groups. We show that the Cauchy problem is well posed. We then select a type of group that allows us to incorporate the experimental information provided by the connection matrices. We argue that the classical Wilson–Cowan model is incompatible with the small-world property. A necessary condition to have this property is that the Wilson–Cowan equations be formulated on a compact group. We propose a p-adic version of the Wilson–Cowan model, a hierarchical version in which the neurons are organized into an infinite rooted tree. We present several numerical simulations showing that the p-adic version matches the predictions of the classical version in relevant experiments. The p-adic version allows the incorporation of the connection matrices into the Wilson–Cowan model. We present several numerical simulations using a neural network model that incorporates a p-adic approximation of the connection matrix of the cat cortex. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
7. Wilson-Cowan Model
- Author
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Kilpatrick, Zachary P., Migliore, Michele, Section editor, Linster, Christiane, Section editor, Cavarretta, Francesco, Section editor, Jaeger, Dieter, editor, and Jung, Ranu, editor
- Published
- 2022
- Full Text
- View/download PDF
8. Noise-modulated multistable synapses in a Wilson-Cowan-based model of plasticity
- Author
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Caroline A. Lea-Carnall, Lisabel I. Tanner, and Marcelo A. Montemurro
- Subjects
Wilson-Cowan model ,plasticity ,multistability ,synapses ,homeostatic ,functional connectivity ,Neurosciences. Biological psychiatry. Neuropsychiatry ,RC321-571 - Abstract
Frequency-dependent plasticity refers to changes in synaptic strength in response to different stimulation frequencies. Resonance is a factor known to be of importance in such frequency dependence, however, the role of neural noise in the process remains elusive. Considering the brain is an inherently noisy system, understanding its effects may prove beneficial in shaping therapeutic interventions based on non-invasive brain stimulation protocols. The Wilson-Cowan (WC) model is a well-established model to describe the average dynamics of neural populations and has been shown to exhibit bistability in the presence of noise. However, the important question of how the different stable regimes in the WC model can affect synaptic plasticity when cortical populations interact has not yet been addressed. Therefore, we investigated plasticity dynamics in a WC-based model of interacting neural populations coupled with activity-dependent synapses in which a periodic stimulation was applied in the presence of noise of controlled intensity. The results indicate that for a narrow range of the noise variance, synaptic strength can be optimized. In particular, there is a regime of noise intensity for which synaptic strength presents a triple-stable state. Regulating noise intensity affects the probability that the system chooses one of the stable states, thereby controlling plasticity. These results suggest that noise is a highly influential factor in determining the outcome of plasticity induced by stimulation.
- Published
- 2023
- Full Text
- View/download PDF
9. Beyond Wilson–Cowan dynamics: oscillations and chaos without inhibition.
- Author
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Painchaud, Vincent, Doyon, Nicolas, and Desrosiers, Patrick
- Subjects
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MARKOV processes , *OSCILLATIONS , *BIOLOGICAL neural networks , *DYNAMICAL systems , *MATHEMATICAL models - Abstract
Fifty years ago, Wilson and Cowan developed a mathematical model to describe the activity of neural populations. In this seminal work, they divided the cells in three groups: active, sensitive and refractory, and obtained a dynamical system to describe the evolution of the average firing rates of the populations. In the present work, we investigate the impact of the often neglected refractory state and show that taking it into account can introduce new dynamics. Starting from a continuous-time Markov chain, we perform a rigorous derivation of a mean-field model that includes the refractory fractions of populations as dynamical variables. Then, we perform bifurcation analysis to explain the occurrence of periodic solutions in cases where the classical Wilson–Cowan does not predict oscillations. We also show that our mean-field model is able to predict chaotic behavior in the dynamics of networks with as little as two populations. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
10. Dynamical Mechanism Analysis of Three Neuroregulatory Strategies on the Modulation of Seizures.
- Author
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Zhang, Honghui, Shen, Zhuan, Zhao, Yuzhi, Du, Lin, and Deng, Zichen
- Subjects
- *
DEEP brain stimulation , *ELECTROMAGNETIC induction , *SUBTHALAMIC nucleus , *SEIZURES (Medicine) , *HOPF bifurcations , *FREQUENCIES of oscillating systems - Abstract
This paper attempts to explore and compare the regulatory mechanisms of optogenetic stimulation (OS), deep brain stimulation (DBS) and electromagnetic induction on epilepsy. Based on the Wilson–Cowan model, we first demonstrate that the external input received by excitatory and inhibitory neural populations can induce rich dynamic bifurcation behaviors such as Hopf bifurcation, and make the system exhibit epileptic and normal states. Then, both OS and DBS are shown to be effective in controlling the epileptic state to a normal low-level state, and the stimulus parameters have a broad effective range. However, electromagnetic induction cannot directly control epilepsy to this desired state, even if it can significantly reduce the oscillation frequency of neural populations. One main difference worth noting is that the high spatiotemporal specificity of OS allows it to target inhibitory neuronal populations, whereas DBS and electromagnetic induction can only stimulate excitatory as well as inhibitory neuronal populations together. Next, the propagation behavior of epilepsy is explored under a typical three-node feedback loop structure. An increase in coupling strength accelerates and exacerbates epileptic activity in other brain regions. Finally, OS and DBS applied to the epileptic focus play similar positive roles in controlling the behavior of the area of seizure propagation, while electromagnetic induction still only achieves unsatisfactory effects. It is hoped that these dynamical results can provide insights into the treatment of epilepsy as well as other neurological disorders. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
11. Oscillations and Synchrony in a Network of Delayed Neural Masses
- Author
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Pinder, Iain, Crofts, Jonathan J., Bandyopadhyay, Anirban, Series Editor, Ray, Kanad, Series Editor, and Poon, Chi-Sang, Series Editor
- Published
- 2021
- Full Text
- View/download PDF
12. Wilson–Cowan Neuronal Interaction Models with Distributed Delays
- Author
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Kaslik, Eva, Kokovics, Emanuel-Attila, Rădulescu, Anca, Lacarbonara, Walter, editor, Balachandran, Balakumar, editor, Ma, Jun, editor, Tenreiro Machado, J. A., editor, and Stepan, Gabor, editor
- Published
- 2020
- Full Text
- View/download PDF
13. Hierarchical Wilson–Cowan Models and Connection Matrices
- Author
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W. A. Zúñiga-Galindo and B. A. Zambrano-Luna
- Subjects
Wilson–Cowan model ,connection matrices ,p-adic numbers ,small-world networks ,Science ,Astrophysics ,QB460-466 ,Physics ,QC1-999 - Abstract
This work aims to study the interplay between the Wilson–Cowan model and connection matrices. These matrices describe cortical neural wiring, while Wilson–Cowan equations provide a dynamical description of neural interaction. We formulate Wilson–Cowan equations on locally compact Abelian groups. We show that the Cauchy problem is well posed. We then select a type of group that allows us to incorporate the experimental information provided by the connection matrices. We argue that the classical Wilson–Cowan model is incompatible with the small-world property. A necessary condition to have this property is that the Wilson–Cowan equations be formulated on a compact group. We propose a p-adic version of the Wilson–Cowan model, a hierarchical version in which the neurons are organized into an infinite rooted tree. We present several numerical simulations showing that the p-adic version matches the predictions of the classical version in relevant experiments. The p-adic version allows the incorporation of the connection matrices into the Wilson–Cowan model. We present several numerical simulations using a neural network model that incorporates a p-adic approximation of the connection matrix of the cat cortex.
- Published
- 2023
- Full Text
- View/download PDF
14. The dance of neurons: Exploring nonlinear dynamics in brain networks.
- Author
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Saadati, Maryam, Khodaei, Saba Sadat, and Jamali, Yousef
- Subjects
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LARGE-scale brain networks , *NEURAL circuitry , *PHASE transitions , *NEURONS , *STIMULUS intensity , *LIMIT cycles - Abstract
The brain is a complex, nonlinear system, exhibiting ever-evolving patterns of activities, whether in the presence or absence of external stimuli or task demands. Nonlinearity can notably obscure the link between structural constraints enforced on the interaction and its dynamical consequences. Suitable nonlinear dynamical models and their analysis serve as essential tools not only for bridging structural and functional understanding of the brain but also for predictably altering the complex dynamical organization of the brain. Here, starting from a large-scale network of threshold Hodgkin–Huxley style neurons, we formulate the average nonlinear dynamics implicitly following from the Wilson–Cowan assumptions. We investigate the influence of biophysical and structural properties on the complexity of neural dynamics at the microscale level and its relationship with the macroscopic Wilson–Cowan model. Incorporating the elements in the model can help identify more realistic regimes of activity and connect the mathematical prediction of increasing nonlinearity to physical manipulations. Our simulations of the temporal profiles reveal dependency on the binary state of interacting subpopulations and the random property of structural network at the transition points, when different synaptic weights are considered. For substantial configurations of stimulus intensity, our model provides further estimates of the neural population's dynamics, ranging from simple-periodic to aperiodic patterns and phase transition regimes. This reflects the potential contribution of the microscopic nonlinear scheme to the mean-field approximation in studying the collective behaviour of individual neurons with particularly concentrating on the occurrence of critical phenomena. We show that finite-size effects kick the system in a state of irregular modes to evolve differently from predictions of the original Wilson–Cowan reference. Additionally, we report that the complexity and temporal diversity of neural dynamics, especially in terms of limit cycle trajectory, and synchronization can be induced by either small heterogeneity in the degree of various types of local excitatory connectivity or considerable diversity in the external drive to the excitatory pool. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
15. Phase response approaches to neural activity models with distributed delay.
- Author
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Winkler, Marius, Dumont, Grégory, Schöll, Eckehard, and Gutkin, Boris
- Subjects
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LIMIT cycles , *LOGNORMAL distribution , *PHASE space , *SYNCHRONIZATION , *COMPUTER simulation , *ADJOINT differential equations - Abstract
In weakly coupled neural oscillator networks describing brain dynamics, the coupling delay is often distributed. We present a theoretical framework to calculate the phase response curve of distributed-delay induced limit cycles with infinite-dimensional phase space. Extending previous works, in which non-delayed or discrete-delay systems were investigated, we develop analytical results for phase response curves of oscillatory systems with distributed delay using Gaussian and log-normal delay distributions. We determine the scalar product and normalization condition for the linearized adjoint of the system required for the calculation of the phase response curve. As a paradigmatic example, we apply our technique to the Wilson–Cowan oscillator model of excitatory and inhibitory neuronal populations under the two delay distributions. We calculate and compare the phase response curves for the Gaussian and log-normal delay distributions. The phase response curves obtained from our adjoint calculations match those compiled by the direct perturbation method, thereby proving that the theory of weakly coupled oscillators can be applied successfully for distributed-delay-induced limit cycles. We further use the obtained phase response curves to derive phase interaction functions and determine the possible phase locked states of multiple inter-coupled populations to illuminate different synchronization scenarios. In numerical simulations, we show that the coupling delay distribution can impact the stability of the synchronization between inter-coupled gamma-oscillatory networks. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
16. Bidirectionally Regulating Gamma Oscillations in Wilson-Cowan Model by Self-Feedback Loops: A Computational Study.
- Author
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Li, XiuPing, Li, ZhengHong, Yang, WanMei, Wu, Zhen, and Wang, JunSong
- Subjects
OSCILLATIONS ,FREQUENCIES of oscillating systems ,SPECTRUM analysis ,COGNITIVE ability - Abstract
The Wilson-Cowan model can emulate gamma oscillations, and thus is extensively used to research the generation of gamma oscillations closely related to cognitive functions. Previous studies have revealed that excitatory and inhibitory inputs to the model can modulate its gamma oscillations. Inhibitory and excitatory self-feedback loops are important structural features of the model, however, its functional role in the regulation of gamma oscillations in the model is still unclear. In the present study, bifurcation analysis and spectrum analysis are employed to elucidate the regulating mechanism of gamma oscillations underlined by the inhibitory and excitatory self-feedback loops, especially how the two self-feedback loops cooperate to generate the gamma oscillations and regulate the oscillation frequency. The present results reveal that, on one hand, the inhibitory self-feedback loop is not conducive to the generation of gamma oscillations, and increased inhibitory self-feedback strength facilitates the enhancement of the oscillation frequency. On the other hand, the excitatory self-feedback loop promotes the generation of gamma oscillations, and increased excitatory self-feedback strength leads to the decrease of oscillation frequency. Finally, theoretical analysis is conducted to provide explain on how the two self-feedback loops play a crucial role in the generation and regulation of neural oscillations in the model. To sum up, Inhibitory and excitatory self-feedback loops play a complementary role in generating and regulating the gamma oscillation in Wilson-Cowan model, and cooperate to bidirectionally regulate the gamma-oscillation frequency in a more flexible manner. These results might provide testable hypotheses for future experimental research. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
17. Bidirectionally Regulating Gamma Oscillations in Wilson-Cowan Model by Self-Feedback Loops: A Computational Study
- Author
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XiuPing Li, ZhengHong Li, WanMei Yang, Zhen Wu, and JunSong Wang
- Subjects
Wilson-Cowan model ,gamma oscillations ,self-feedback loops ,bifurcation analysis ,spectrum analysis ,Neurosciences. Biological psychiatry. Neuropsychiatry ,RC321-571 - Abstract
The Wilson-Cowan model can emulate gamma oscillations, and thus is extensively used to research the generation of gamma oscillations closely related to cognitive functions. Previous studies have revealed that excitatory and inhibitory inputs to the model can modulate its gamma oscillations. Inhibitory and excitatory self-feedback loops are important structural features of the model, however, its functional role in the regulation of gamma oscillations in the model is still unclear. In the present study, bifurcation analysis and spectrum analysis are employed to elucidate the regulating mechanism of gamma oscillations underlined by the inhibitory and excitatory self-feedback loops, especially how the two self-feedback loops cooperate to generate the gamma oscillations and regulate the oscillation frequency. The present results reveal that, on one hand, the inhibitory self-feedback loop is not conducive to the generation of gamma oscillations, and increased inhibitory self-feedback strength facilitates the enhancement of the oscillation frequency. On the other hand, the excitatory self-feedback loop promotes the generation of gamma oscillations, and increased excitatory self-feedback strength leads to the decrease of oscillation frequency. Finally, theoretical analysis is conducted to provide explain on how the two self-feedback loops play a crucial role in the generation and regulation of neural oscillations in the model. To sum up, Inhibitory and excitatory self-feedback loops play a complementary role in generating and regulating the gamma oscillation in Wilson-Cowan model, and cooperate to bidirectionally regulate the gamma-oscillation frequency in a more flexible manner. These results might provide testable hypotheses for future experimental research.
- Published
- 2022
- Full Text
- View/download PDF
18. Dynamical Mechanism Analysis of Three Neuroregulatory Strategies on the Modulation of Seizures
- Author
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Honghui Zhang, Zhuan Shen, Yuzhi Zhao, Lin Du, and Zichen Deng
- Subjects
Wilson–Cowan model ,optogenetic stimulation ,deep brain stimulation ,electromagnetic induction ,epilepsy ,Biology (General) ,QH301-705.5 ,Chemistry ,QD1-999 - Abstract
This paper attempts to explore and compare the regulatory mechanisms of optogenetic stimulation (OS), deep brain stimulation (DBS) and electromagnetic induction on epilepsy. Based on the Wilson–Cowan model, we first demonstrate that the external input received by excitatory and inhibitory neural populations can induce rich dynamic bifurcation behaviors such as Hopf bifurcation, and make the system exhibit epileptic and normal states. Then, both OS and DBS are shown to be effective in controlling the epileptic state to a normal low-level state, and the stimulus parameters have a broad effective range. However, electromagnetic induction cannot directly control epilepsy to this desired state, even if it can significantly reduce the oscillation frequency of neural populations. One main difference worth noting is that the high spatiotemporal specificity of OS allows it to target inhibitory neuronal populations, whereas DBS and electromagnetic induction can only stimulate excitatory as well as inhibitory neuronal populations together. Next, the propagation behavior of epilepsy is explored under a typical three-node feedback loop structure. An increase in coupling strength accelerates and exacerbates epileptic activity in other brain regions. Finally, OS and DBS applied to the epileptic focus play similar positive roles in controlling the behavior of the area of seizure propagation, while electromagnetic induction still only achieves unsatisfactory effects. It is hoped that these dynamical results can provide insights into the treatment of epilepsy as well as other neurological disorders.
- Published
- 2022
- Full Text
- View/download PDF
19. Emergence of synchronised and amplified oscillations in neuromorphic networks with long-range interactions.
- Author
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Apicella, I., Busiello, D.M., Scarpetta, S., and Suweis, S.
- Subjects
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OSCILLATIONS , *DISTRIBUTION (Probability theory) , *STOCHASTIC resonance , *INFERIOR colliculus , *HETEROGENEITY - Abstract
• We characterise effects of parameter heterogeneity in a neuromorphic directed chain. • Long-range links trigger the emergence of synchronisation and phase coherence. • Local topological condition leads to strong amplification even with backward links. Neuromorphic networks can be described in terms of coarse-grained variables, where emergent sustained behaviours spontaneously arise if stochasticity is properly taken into account. For example it has been recently found that a directed linear chain of connected patch of neurons amplifies an input signal, also tuning its characteristic frequency. Here we study a generalization of such a simple model, introducing heterogeneity and variability in the parameter space and long-range interactions, breaking, in turn, the preferential direction of information transmission of a directed chain. On one hand, enlarging the region of parameters leads to a more complex state space that we analytically characterise; moreover, we explicitly link the strength distribution of the non-local interactions with the frequency distribution of the network oscillations. On the other hand, we found that adding long-range interactions can cause the onset of novel phenomena, as coherent and synchronous oscillations among all the interacting units, which can also coexist with the amplification of the signal. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
20. Bifurcations in a forced Wilson-Cowan neuron pair
- Author
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Masaki Yoshikawa, Kentaro Ono, and Tetsushi Ueta
- Subjects
Wilson-Cowan model ,Polymers and Plastics ,brute-force analysis ,chaos ,bifurcation ,entrainment ,torus ,General Environmental Science - Abstract
We investigate bifurcations of periodic solutions observed in the forced Wilson-Cowan neuron pair by both the brute-force computation and the shooting method. By superimposing the results given by both methods, a detailed topological classification of periodic solutions is achieved that includes tori and chaos attractors in the parameter space is achieved. We thoroughly explore the parameter space composed of threshold values, amplitude, and angular velocity of an external forcing term. Many bifurcation curves that are invisible when using brute-force method are solved by the shooting method. We find out a typical bifurcation structure including Arnold tongue in the angular velocity and the amplitude of the external force parameter plane, and confirm its fractal structure. In addition, the emergence of periodic bursting responses depending on these patterns is explained.
- Published
- 2023
21. Divisive gain modulation enables flexible and rapid entrainment in a neocortical microcircuit model.
- Author
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Papasavvas, Christoforos A., Trevelyan, Andrew J., Kaiser, Marcus, and Yujiang Wang
- Abstract
Neocortical circuits exhibit a rich dynamic repertoire, and their ability to achieve entrainment (adjustment of their frequency to match the input frequency) is thought to support many cognitive functions and indicate functional flexibility. Although previous studies have explored the influence of various circuit properties on this phenomenon, the role of divisive gain modulation (or divisive inhibition) is unknown. This gain control mechanism is thought to be delivered mainly by the soma-targeting interneurons in neocortical microcircuits. In this study, we use a neural mass model of the neocortical microcircuit (extended Wilson–Cowan model) featuring both soma-targeting and dendritetargeting interneuronal subpopulations to investigate the role of divisive gain modulation in entrainment. Our results demonstrate that the presence of divisive inhibition in the microcircuit, as delivered by the soma-targeting interneurons, enables its entrainment to a wider range of input frequencies. Divisive inhibition also promotes a faster entrainment, with the microcircuit needing less time to converge to the fully entrained state. We suggest that divisive inhibition, working alongside subtractive inhibition, allows for more adaptive oscillatory responses in neocortical circuits and, thus, supports healthy brain functioning. NEW & NOTEWORTHY We introduce a computational neocortical microcircuit model that features two inhibitory neural populations, with one providing subtractive and the other divisive inhibition to the excitatory population. We demonstrate that divisive inhibition widens the range of input frequencies to which the microcircuit can become entrained and diminishes the time needed to reach full entrainment. We suggest that divisive inhibition enables more adaptive oscillatory activity, with important implications for both normal and pathological brain function. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
22. Bifurcations in a forced Wilson-Cowan neuron pair
- Author
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Yoshikawa, Masaki, Ono, Kentaro, Ueta, Tetsushi, Yoshikawa, Masaki, Ono, Kentaro, and Ueta, Tetsushi
- Abstract
We investigate bifurcations of periodic solutions observed in the forced Wilson-Cowan neuron pair by both the brute-force computation and the shooting method. By superimposing the results given by both methods, a detailed topological classification of periodic solutions is achieved that includes tori and chaos attractors in the parameter space is achieved. We thoroughly explore the parameter space composed of threshold values, amplitude, and angular velocity of an external forcing term. Many bifurcation curves that are invisible when using brute-force method are solved by the shooting method. We find out a typical bifurcation structure including Arnold tongue in the angular velocity and the amplitude of the external force parameter plane, and confirm its fractal structure. In addition, the emergence of periodic bursting responses depending on these patterns is explained.
- Published
- 2023
23. Effects of optogenetic and visual stimulation on gamma activity in the visual cortex.
- Author
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Arab, Fereshteh, Rostami, Sareh, Dehghani-Habibabadi, Mohammad, Mateos, Diego M., Braddell, Roisin, Scholkmann, Felix, Ismail Zibaii, Mohammad, Rodrigues, Serafim, Salari, Vahid, and Safari, Mir-Shahram
- Subjects
- *
VISUAL cortex , *VISUAL perception , *FREQUENCIES of oscillating systems , *REINFORCEMENT (Psychology) , *OSCILLATIONS , *DATA analysis - Abstract
• Feasibility of conducting cognitive experiments during brain gamma oscillations. • Optogenetic stimulation enhances low gamma power in the visual cortex. • Visual stimulation has differing effects on gamma power in different layers of the visual cortex. Studying brain functions and activity during gamma oscillations can be a challenge because it requires careful planning to create the necessary conditions for a controlled experiment. Such an experiment consists of placing the brain into a gamma state and investigating cognitive processing with a careful design. Cortical oscillations in the gamma frequency range (30–80 Hz) play an essential role in a variety of cognitive processes, including visual processing and cognition. The present study aims to investigate the effects of a visual stimulus on the primary visual cortex under gamma oscillations. Specifically, we sought to explore the behavior of gamma oscillations triggered by optogenetic stimulation in the II and IV layers of the visual cortex, both with and without concurrent visual stimulation. Our results show that optogenetic stimulation increases the power of gamma oscillation in both layers of the visual cortex. However, the combined stimuli resulted in a reduction of gamma power in layer II and an increase and reinforcement in gamma power in layer IV. Modelling the results with the Wilson-Cowan model suggests changes in the input of the excitatory population due to the combined stimuli. In addition, our analysis of the data using the Lempel-Ziv complexity method supports our interpretations from the modeling. Thus, our results suggest that optogenetic stimulation enhances low gamma power in both layers of the visual cortex, while simultaneous visual stimulation has differing effects on the two layers, reducing gamma power in layer II and increasing it in layer IV. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
24. Hierarchical Wilson–Cowan Models and Connection Matrices
- Author
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Zambrano-Luna, W. A. Zúñiga-Galindo and B. A.
- Subjects
Wilson–Cowan model ,connection matrices ,p-adic numbers ,small-world networks - Abstract
This work aims to study the interplay between the Wilson–Cowan model and connection matrices. These matrices describe cortical neural wiring, while Wilson–Cowan equations provide a dynamical description of neural interaction. We formulate Wilson–Cowan equations on locally compact Abelian groups. We show that the Cauchy problem is well posed. We then select a type of group that allows us to incorporate the experimental information provided by the connection matrices. We argue that the classical Wilson–Cowan model is incompatible with the small-world property. A necessary condition to have this property is that the Wilson–Cowan equations be formulated on a compact group. We propose a p-adic version of the Wilson–Cowan model, a hierarchical version in which the neurons are organized into an infinite rooted tree. We present several numerical simulations showing that the p-adic version matches the predictions of the classical version in relevant experiments. The p-adic version allows the incorporation of the connection matrices into the Wilson–Cowan model. We present several numerical simulations using a neural network model that incorporates a p-adic approximation of the connection matrix of the cat cortex.
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- 2023
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25. Wilson-Cowan Model
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Kilpatrick, Zachary P., Jaeger, Dieter, editor, and Jung, Ranu, editor
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- 2015
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26. Noise-modulated multistable synapses in a Wilson-Cowan-based model of plasticity
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Lea-Carnall, Caroline A., Tanner, Lisabel I., and Montemurro, Marcelo A.
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Cellular and Molecular Neuroscience ,Wilson-Cowan model ,multistability ,plasticity ,functional connectivity ,homeostatic ,Neuroscience (miscellaneous) ,synapses ,neural mass model - Abstract
Frequency-dependent plasticity refers to changes in synaptic strength in response to different stimulation frequencies. Resonance is a factor known to be of importance in such frequency dependence, however, the role of neural noise in the process remains elusive. Considering the brain is an inherently noisy system, understanding its effects may prove beneficial in shaping therapeutic interventions based on non-invasive brain stimulation protocols. The Wilson-Cowan (WC) model is a well-established model to describe the average dynamics of neural populations and has been shown to exhibit bistability in the presence of noise. However, the important question of how the different stable regimes in the WC model can affect synaptic plasticity when cortical populations interact has not yet been addressed. Therefore, we investigated plasticity dynamics in a WC-based model of interacting neural populations coupled with activity-dependent synapses in which a periodic stimulation was applied in the presence of noise of controlled intensity. The results indicate that for a narrow range of the noise variance, synaptic strength can be optimized. In particular, there is a regime of noise intensity for which synaptic strength presents a triple-stable state. Regulating noise intensity affects the probability that the system chooses one of the stable states, thereby controlling plasticity. These results suggest that noise is a highly influential factor in determining the outcome of plasticity induced by stimulation.
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- 2023
27. Firing rate models for gamma oscillations.
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Keeley, Stephen, Byrne, Áine, Fenton, André, and Rinzel, John
- Abstract
Gamma oscillations are readily observed in a variety of brain regions during both waking and sleeping states. Computational models of gamma oscillations typically involve simulations of large networks of synaptically coupled spiking units. These networks can exhibit strongly synchronized gamma behavior, whereby neurons fire in near synchrony on every cycle, or weakly modulated gamma behavior, corresponding to stochastic, sparse firing of the individual units on each cycle of the population gamma rhythm. These spiking models offer valuable biophysical descriptions of gamma oscillations; however, because they involve many individual neuronal units they are limited in their ability to communicate general network-level dynamics. Here we demonstrate that few-variable firing rate models with established synaptic timescales can account for both strongly synchronized and weakly modulated gamma oscillations. These models go beyond the classical formulations of rate models by including at least two dynamic variables per population: firing rate and synaptic activation. The models’ flexibility to capture the broad range of gamma behavior depends directly on the timescales that represent recruitment of the excitatory and inhibitory firing rates. In particular, we find that weakly modulated gamma oscillations occur robustly when the recruitment timescale of inhibition is faster than that of excitation. We present our findings by using an extended Wilson-Cowan model and a rate model derived from a network of quadratic integrate-and-fire neurons. These biophysical rate models capture the range of weakly modulated and coherent gamma oscillations observed in spiking network models, while additionally allowing for greater tractability and systems analysis. [ABSTRACT FROM AUTHOR]
- Published
- 2019
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28. Mean field dynamics of a Wilson–Cowan neuronal network with nonlinear coupling term.
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MacLaurin, James, Salhi, Jamil, and Toumi, Salwa
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- *
MEAN field theory , *BROWNIAN motion , *RANDOM noise theory , *STOCHASTIC differential equations , *MAGNETIC coupling - Abstract
In this paper we prove the propagation of chaos property for an ensemble of interacting neurons subject to independent Brownian noise. The propagation of chaos property means that in the large network size limit, the neurons behave as if they are probabilistically independent. The model for the internal dynamics of the neurons is taken to be that of Wilson and Cowan, and we consider there to be multiple different populations. The synaptic connections are modeled with a nonlinear "electrical" model. The nonlinearity of the synaptic connections means that our model lies outside the scope of classical propagation of chaos results. We obtain the propagation of chaos result by taking advantage of the fact that the mean-field equations are Gaussian, which allows us to use Borell's Inequality to prove that its tails decay exponentially. [ABSTRACT FROM AUTHOR]
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- 2018
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29. Phase response approaches to neural activity models with distributed delay
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Boris Gutkin, Eckehard Schöll, Marius Winkler, and Grégory Dumont
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Neurons ,Physics ,General Computer Science ,Gaussian ,Scalar (physics) ,Phase (waves) ,Brain ,Stability (probability) ,Wilson–Cowan model ,symbols.namesake ,Phase space ,Phase response ,symbols ,Neural Networks, Computer ,Statistical physics ,Biotechnology ,Phase response curve - Abstract
In weakly coupled neural oscillator networks describing brain dynamics, the coupling delay is often distributed. We present a theoretical framework to calculate the phase response curve of distributed-delay induced limit cycles with infinite-dimensional phase space. Extending previous works, in which non-delayed or discrete-delay systems were investigated, we develop analytical results for phase response curves of oscillatory systems with distributed delay using Gaussian and log-normal delay distributions. We determine the scalar product and normalization condition for the linearized adjoint of the system required for the calculation of the phase response curve. As a paradigmatic example, we apply our technique to the Wilson-Cowan oscillator model of excitatory and inhibitory neuronal populations under the two delay distributions. We calculate and compare the phase response curves for the Gaussian and log-normal delay distributions. The phase response curves obtained from our adjoint calculations match those compiled by the direct perturbation method, thereby proving that the theory of weakly coupled oscillators can be applied successfully for distributed-delay-induced limit cycles. We further use the obtained phase response curves to derive phase interaction functions and determine the possible phase locked states of multiple inter-coupled populations to illuminate different synchronization scenarios. In numerical simulations, we show that the coupling delay distribution can impact the stability of the synchronization between inter-coupled gamma-oscillatory networks.
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- 2021
30. Temporal characteristics of gamma rhythm constrain properties of noise in an inhibition-stabilized network model.
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Krishnakumaran R and Ray S
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- Noise, Gamma Rhythm physiology, Visual Cortex physiology
- Abstract
Gamma rhythm refers to oscillatory neural activity between 30 and 80 Hz, induced in visual cortex by stimuli such as iso-luminant hues or gratings. The power and peak frequency of gamma depend on the properties of the stimulus such as size and contrast. Gamma waveform is typically arch-shaped, with narrow troughs and broad peaks, and can be replicated in a self-oscillating Wilson-Cowan (WC) model operating in an appropriate regime. However, oscillations in this model are infinitely long, unlike physiological gamma that occurs in short bursts. Further, unlike the model, gamma is faster after stimulus onset and slows down over time. Here, we first characterized gamma burst duration in local field potential data recorded from two monkeys as they viewed full screen iso-luminant hues. We then added different types of noise in the inputs to the WC model and tested how that affected duration and temporal dynamics of gamma. While the model failed with the often-used Poisson noise, Ornstein-Uhlenbeck noise applied to both the excitatory and the inhibitory populations replicated the duration and slowing of gamma and replicated the shape and stimulus dependencies. Thus, the temporal dynamics of gamma oscillations put constraints on the type and properties of underlying neural noise., (© The Author(s) 2023. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com.)
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- 2023
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31. Scale-freeness or partial synchronization in neural mass phase oscillator networks: Pick one of two?
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Daffertshofer, Andreas, Ton, Robert, Pietras, Bastian, Deco, Gustavo, and Kringelbach, Morten L.
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- *
SYNCHRONIZATION , *K-means clustering , *OSCILLATOR strengths , *AUTOCORRELATION (Statistics) , *DYNAMICS - Abstract
Abstract Modeling and interpreting (partial) synchronous neural activity can be a challenge. We illustrate this by deriving the phase dynamics of two seminal neural mass models: the Wilson-Cowan firing rate model and the voltage-based Freeman model. We established that the phase dynamics of these models differed qualitatively due to an attractive coupling in the first and a repulsive coupling in the latter. Using empirical structural connectivity matrices, we determined that the two dynamics cover the functional connectivity observed in resting state activity. We further searched for two pivotal dynamical features that have been reported in many experimental studies: (1) a partial phase synchrony with a possibility of a transition towards either a desynchronized or a (fully) synchronized state; (2) long-term autocorrelations indicative of a scale-free temporal dynamics of phase synchronization. Only the Freeman phase model exhibited scale-free behavior. Its repulsive coupling, however, let the individual phases disperse and did not allow for a transition into a synchronized state. The Wilson-Cowan phase model, by contrast, could switch into a (partially) synchronized state, but it did not generate long-term correlations although being located close to the onset of synchronization, i.e. in its critical regime. That is, the phase-reduced models can display one of the two dynamical features, but not both. Highlights • Networks of neural masses are reduced to phase oscillator models. • Freeman- and Wilson-Cowan-based phase models can resemble empirically observed functional connectivity. • Neither a network of Freeman nor of Wilson-Cowan phase models can. • Capture both partial synchronization and scale-free temporal correlations. [ABSTRACT FROM AUTHOR]
- Published
- 2018
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32. To swim or not to swim: A population-level model of Xenopus tadpole decision making and locomotor behaviour.
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Borisyuk, Roman, Merrison-Hort, Robert, Soffe, Steve R., Koutsikou, Stella, and Li, Wen-Chang
- Subjects
- *
FISH locomotion , *XENOPUS , *FISHES , *CENTRAL pattern generators , *DECISION making , *BEHAVIOR , *MOTOR neurons - Abstract
We present a detailed computational model of interacting neuronal populations that mimic the hatchling Xenopus tadpole nervous system. The model includes four sensory pathways, integrators of sensory information, and a central pattern generator (CPG) network. Sensory pathways of different modalities receive inputs from an “environment”; these inputs are then processed and integrated to select the most appropriate locomotor action. The CPG populations execute the selected action, generating output in motor neuron populations. Thus, the model describes a detailed and biologically plausible chain of information processing from external signals to sensors, sensory pathways, integration and decision-making, action selection and execution and finally, generation of appropriate motor activity and behaviour. We show how the model produces appropriate behaviours in response to a selected scenario, which consists of a sequence of “environmental” signals. These behaviours might be relatively complex due to noisy sensory pathways and the possibility of spontaneous actions. [ABSTRACT FROM AUTHOR]
- Published
- 2017
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33. Bifurcation Analysis of Wilson-Cowan Model with Singular Impulses
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Marat Akhmet and Sabahattin Çağ
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Physics ,Control and Optimization ,Mathematical analysis ,Computational Mechanics ,Structure (category theory) ,Novelty ,Statistical and Nonlinear Physics ,Dynamical Systems (math.DS) ,Wilson–Cowan model ,Bifurcation analysis ,Singularity ,Phase space ,Attractor ,FOS: Mathematics ,Discrete Mathematics and Combinatorics ,Mathematics - Dynamical Systems ,Bifurcation - Abstract
The paper concerns with Wilson-Cowan neural model with impulses. The main novelty of the study is that besides the traditional singularity of the model, we consider singular impulses. A new technique of analysis of the phenomenon is suggested. This allows to consider the existence of solutions of the model and bifurcation in ultimate neural behavior is observed through numerical simulations. The bifurcations are reasoned by impulses and singularity in the model and they concern the structure of attractors, which consist of newly introduced sets in the phase space such that medusas and rings., 15 pages, 9 figures
- Published
- 2021
34. The influence of depolarization block on seizure-like activity in networks of excitatory and inhibitory neurons.
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Kim, Christopher and Nykamp, Duane
- Abstract
The inhibitory restraint necessary to suppress aberrant activity can fail when inhibitory neurons cease to generate action potentials as they enter depolarization block. We investigate possible bifurcation structures that arise at the onset of seizure-like activity resulting from depolarization block in inhibitory neurons. Networks of conductance-based excitatory and inhibitory neurons are simulated to characterize different types of transitions to the seizure state, and a mean field model is developed to verify the generality of the observed phenomena of excitatory-inhibitory dynamics. Specifically, the inhibitory population's activation function in the Wilson-Cowan model is modified to be non-monotonic to reflect that inhibitory neurons enter depolarization block given strong input. We find that a physiological state and a seizure state can coexist, where the seizure state is characterized by high excitatory and low inhibitory firing rate. Bifurcation analysis of the mean field model reveals that a transition to the seizure state may occur via a saddle-node bifurcation or a homoclinic bifurcation. We explain the hysteresis observed in network simulations using these two bifurcation types. We also demonstrate that extracellular potassium concentration affects the depolarization block threshold; the consequent changes in bifurcation structure enable the network to produce the tonic to clonic phase transition observed in biological epileptic networks. [ABSTRACT FROM AUTHOR]
- Published
- 2017
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35. Connectome-based prediction of functional impairment in experimental stroke models.
- Author
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Schmitt O, Eipert P, Wang Y, Kanoke A, Rabiller G, and Liu J
- Abstract
Experimental rat models of stroke and hemorrhage are important tools to investigate cerebrovascular disease pathophysiology mechanisms, yet how significant patterns of functional impairment induced in various models of stroke are related to changes in connectivity at the level of neuronal populations and mesoscopic parcellations of rat brains remain unresolved. To address this gap in knowledge, we employed two middle cerebral artery occlusion models and one intracerebral hemorrhage model with variant extent and location of neuronal dysfunction. Motor and spatial memory function was assessed and the level of hippocampal activation via Fos immunohistochemistry. Contribution of connectivity change to functional impairment was analyzed for connection similarities, graph distances and spatial distances as well as the importance of regions in terms of network architecture based on the neuroVIISAS rat connectome. We found that functional impairment correlated with not only the extent but also the locations of the injury among the models. In addition, via coactivation analysis in dynamic rat brain models, we found that lesioned regions led to stronger coactivations with motor function and spatial learning regions than with other unaffected regions of the connectome. Dynamic modeling with the weighted bilateral connectome detected changes in signal propagation in the remote hippocampus in all 3 stroke types, predicting the extent of hippocampal hypoactivation and impairment in spatial learning and memory function. Our study provides a comprehensive analytical framework in predictive identification of remote regions not directly altered by stroke events and their functional implication., Competing Interests: Disclosure/Conflict of Interest The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
- Published
- 2023
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36. Divisive gain modulation enables flexible and rapid entrainment in a neocortical microcircuit model
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Andrew J. Trevelyan, Marcus Kaiser, Christoforos A. Papasavvas, and Yujiang Wang
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Physiology ,Models, Neurological ,Neocortex ,neural entrainment ,Inhibitory postsynaptic potential ,03 medical and health sciences ,gain control ,0302 clinical medicine ,Interneurons ,Animals ,Humans ,030304 developmental biology ,Physics ,0303 health sciences ,musculoskeletal, neural, and ocular physiology ,General Neuroscience ,Neural Inhibition ,inhibitory interneurons ,Wilson–Cowan model ,Excitatory postsynaptic potential ,Neural Networks, Computer ,Nerve Net ,Entrainment (chronobiology) ,Neuroscience ,030217 neurology & neurosurgery ,Research Article ,neocortical dynamics - Abstract
Neocortical circuits exhibit a rich dynamic repertoire, and their ability to achieve entrainment (adjustment of their frequency to match the input frequency) is thought to support many cognitive functions and indicate functional flexibility. Although previous studies have explored the influence of various circuit properties on this phenomenon, the role of divisive gain modulation (or divisive inhibition) is unknown. This gain control mechanism is thought to be delivered mainly by the soma-targeting interneurons in neocortical microcircuits. In this study, we use a neural mass model of the neocortical microcircuit (extended Wilson–Cowan model) featuring both soma-targeting and dendrite-targeting interneuronal subpopulations to investigate the role of divisive gain modulation in entrainment. Our results demonstrate that the presence of divisive inhibition in the microcircuit, as delivered by the soma-targeting interneurons, enables its entrainment to a wider range of input frequencies. Divisive inhibition also promotes a faster entrainment, with the microcircuit needing less time to converge to the fully entrained state. We suggest that divisive inhibition, working alongside subtractive inhibition, allows for more adaptive oscillatory responses in neocortical circuits and, thus, supports healthy brain functioning. NEW & NOTEWORTHY We introduce a computational neocortical microcircuit model that features two inhibitory neural populations, with one providing subtractive and the other divisive inhibition to the excitatory population. We demonstrate that divisive inhibition widens the range of input frequencies to which the microcircuit can become entrained and diminishes the time needed to reach full entrainment. We suggest that divisive inhibition enables more adaptive oscillatory activity, with important implications for both normal and pathological brain function.
- Published
- 2020
37. Critical behaviour of the stochastic Wilson-Cowan model
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Lucilla de Arcangelis, Antonio de Candia, A. Sarracino, Ilenia Apicella, De Candia, A, Sarracino, A., Apicella, I., de Arcangelis, L., de Candia, Antonio, Sarracino, Alessandro, Apicella, Ilenia, and de Arcangelis, Lucilla
- Subjects
Phase transition ,Physiology ,Social Sciences ,Action Potentials ,Parameter space ,Critical point (thermodynamics) ,Animal Cells ,Relaxation Time ,Psychology ,Statistical physics ,Biology (General) ,Physics ,Neurons ,Ecology ,Applied Mathematics ,Simulation and Modeling ,Brain ,Electrophysiology ,Computational Theory and Mathematics ,Modeling and Simulation ,Physical Sciences ,Thermodynamics ,Cellular Types ,Algorithms ,Neuronal Tuning ,Research Article ,Computer and Information Sciences ,Neural Networks ,QH301-705.5 ,Models, Neurological ,Neurophysiology ,Research and Analysis Methods ,Membrane Potential ,Cellular and Molecular Neuroscience ,Synaptic weight ,Spatio-Temporal Analysis ,Genetics ,Animals ,Humans ,Computer Simulation ,Molecular Biology ,Scaling ,Relaxation (Physics) ,Ecology, Evolution, Behavior and Systematics ,Branching process ,Behavior ,Stochastic Processes ,Quantitative Biology::Neurons and Cognition ,Biology and Life Sciences ,Computational Biology ,Cell Biology ,Renormalization group ,Wilson–Cowan model ,Electrophysiological Phenomena ,Cellular Neuroscience ,Linear Models ,Linear approximation ,Nerve Net ,Mathematics ,Neuroscience - Abstract
Spontaneous brain activity is characterized by bursts and avalanche-like dynamics, with scale-free features typical of critical behaviour. The stochastic version of the celebrated Wilson-Cowan model has been widely studied as a system of spiking neurons reproducing non-trivial features of the neural activity, from avalanche dynamics to oscillatory behaviours. However, to what extent such phenomena are related to the presence of a genuine critical point remains elusive. Here we address this central issue, providing analytical results in the linear approximation and extensive numerical analysis. In particular, we present results supporting the existence of a bona fide critical point, where a second-order-like phase transition occurs, characterized by scale-free avalanche dynamics, scaling with the system size and a diverging relaxation time-scale. Moreover, our study shows that the observed critical behaviour falls within the universality class of the mean-field branching process, where the exponents of the avalanche size and duration distributions are, respectively, 3/2 and 2. We also provide an accurate analysis of the system behaviour as a function of the total number of neurons, focusing on the time correlation functions of the firing rate in a wide range of the parameter space., Author summary Networks of spiking neurons are introduced to describe some features of the brain activity, which are characterized by burst events (avalanches) with power-law distributions of size and duration. The observation of this kind of noisy behaviour in a wide variety of real systems led to the hypothesis that neuronal networks work in the proximity of a critical point. This hypothesis is at the core of an intense debate. At variance with previous claims, here we show that a stochastic version of the Wilson-Cowan model presents a phenomenology in agreement with the existence of a bona fide critical point for a particular choice of the relative synaptic weight between excitatory and inhibitory neurons. The system behaviour at this point shows all features typical of criticality, such as diverging timescales, scaling with the system size and scale-free distributions of avalanche sizes and durations, with exponents corresponding to the mean-field branching process. Our analysis unveils the critical nature of the observed behaviours.
- Published
- 2021
38. The role of network structure and time delay in a metapopulation Wilson--Cowan model
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Federica Conti and Robert A. Van Gorder
- Subjects
0301 basic medicine ,Statistics and Probability ,Computer science ,Population ,Network topology ,Topology ,Models, Biological ,General Biochemistry, Genetics and Molecular Biology ,Synchronization ,03 medical and health sciences ,0302 clinical medicine ,Limit cycle ,education ,education.field_of_study ,Quantitative Biology::Neurons and Cognition ,General Immunology and Microbiology ,Applied Mathematics ,General Medicine ,Delay differential equation ,Degree distribution ,Wilson–Cowan model ,030104 developmental biology ,Modeling and Simulation ,Node (circuits) ,General Agricultural and Biological Sciences ,030217 neurology & neurosurgery - Abstract
We study the dynamics of a network Wilson--Cowan model (a system of connected Wilson--Cowan oscillators) for interacting excitatory and inhibitory neuron populations with time delays. Each node in this model corresponds to a population of neurons, including excitatory and inhibitory subpopulations, and hence it can be viewed as a metapopulation model. It is known that information transfer within each cortical area is not instantaneous, and therefore we consider a system of delay differential equations with two different kinds of discrete delays. We account for the time delay in information propagation between individual excitatory and inhibitory subpopulations at each node via intra-node time delays, and we account for time delay in information propagation between neuron populations at different nodes with inter-node time delays. The biologically relevant resting state solutions are oscillatory (stable limit cycles). After determining the influence of the coupling parameters between nodes, the intra-node delays, and the inter-node delays on the dynamics of the two coupled Wilson--Cowan oscillators, we then explore a variety of larger networks of 16 and 100 nodes, in order to determine how the network topology will influence time delayed Wilson--Cowan dynamics. We find that network structure can regularize or deregularize the dynamics, with networks of higher mean degree permitting stable limit cycles and networks with smaller mean degree yielding less regular dynamics (which may range from chaotic solutions, to solutions for which limit cycles collapse into steady states, which are biologically undesirable compared with the preferred stable limit cycles). Furthermore, heterogeneity in the degree distribution of the network (resulting from networks with nodes of varying degree) can result in asynchronous dynamics, even if at each node the local dynamics are that of a limit cycle, in contrast to the synchronization of dynamics between nodes seen when the degree of all nodes is equal. This suggests that homogeneous and well-connected networks permit robust limit cycles under time-delayed Wilson--Cowan dynamics, whereas heterogeneous or poorly connected networks may fail to provide such desirable dynamics, a phenomena akin to structural loss of neuron connections in neurodegenerative diseases.
- Published
- 2019
39. Generalized activity equations for spiking neural network dynamics
- Author
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Michael A Buice and Carson C Chow
- Subjects
fluctuations ,correlations ,Fokker-Planck ,Wilson-Cowan model ,Finite size networks ,mean field theory ,Neurosciences. Biological psychiatry. Neuropsychiatry ,RC321-571 - Abstract
Much progress has been made in uncovering the computational capabilities of spiking neural networks. However, spiking neurons will always be more expensive to simulate compared to rate neurons because of the inherent disparity in time scales - the spike duration time is much shorter than the inter-spike time, which is much shorter than any learning time scale. In numerical analysis, this is a classic stiff problem. Spiking neurons are also much more difficult to study analytically. One possible approach to making spiking networks more tractable is to augment mean field activity models with some information about spiking correlations. For example, such a generalized activity model could carry information about spiking rates and correlations between spikes self-consistently. Here, we will show how this can be accomplished by constructing a complete formal probabilistic description of the network and then expanding around a small parameter such as the inverse of the number of neurons in the network. The mean field theory of the system gives a rate-like description. The first order terms in the perturbation expansion keep track of covariances.
- Published
- 2013
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40. Noise-Induced Precursors of State Transitions in the Stochastic Wilson-Cowan Model.
- Author
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Negahbani, Ehsan, Steyn-Ross, D. Alistair, Steyn-Ross, Moira L., Wilson, Marcus T., and Sleigh, Jamie W.
- Subjects
- *
BRAIN physiology , *NEURAL physiology , *BEHAVIORAL assessment , *STOCHASTIC analysis , *DIFFERENTIAL equations , *FEASIBILITY studies , *COMPUTER simulation - Abstract
The Wilson-Cowan neural field equations describe the dynamical behavior of a 1-D continuum of excitatory and inhibitory cortical neural aggregates, using a pair of coupled integro-differential equations. Here we use bifurcation theory and small-noise linear stochastics to study the range of a phase transitions--sudden qualitative changes in the state of a dynamical system emerging from a bifurcation-- accessible to the Wilson-Cowan network. Specifically, we examine saddle-node, Hopf, Turing, and Turing-Hopf instabilities. We introduce stochasticity by adding small-amplitude spatio-temporal white noise, and analyze the resulting subthreshold fluctuations using an Ornstein-Uhlenbeck linearization. This analysis predicts divergent changes in correlation and spectral characteristics of neural activity during close approach to bifurcation from below. We validate these theoretical predictions using numerical simulations. The results demonstrate the role of noise in the emergence of critically slowed precursors in both space and time, and suggest that these earlywarning signals are a universal feature of a neural system close to bifurcation. In particular, these precursor signals are likely to have neurobiological significance as early warnings of impending state change in the cortex. We support this claim with an analysis of the in vitro local field potentials recorded from slices of mouse-brain tissue. We show that in the period leading up to emergence of spontaneous seizurelike events, the mouse field potentials show a characteristic spectral focusing toward lower frequencies concomitant with a growth in fluctuation variance, consistent with critical slowing near a bifurcation point. This observation of biological criticality has clear implications regarding the feasibility of seizure prediction. [ABSTRACT FROM AUTHOR]
- Published
- 2015
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41. Analysis of chaotic oscillations induced in two coupled Wilson-Cowan models.
- Author
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Maruyama, Yuya, Kakimoto, Yuta, and Araki, Osamu
- Subjects
- *
CHAOS theory , *OSCILLATIONS , *DYNAMICAL systems , *BIFURCATION theory , *MATHEMATICAL models - Abstract
Although it is known that two coupled Wilson-Cowan models with reciprocal connections induce aperiodic oscillations, little attention has been paid to the dynamical mechanism for such oscillations so far. In this study, we aim to elucidate the fundamental mechanism to induce the aperiodic oscillations in the coupled model. First, aperiodic oscillations observed are investigated for the case when the connections are unidirectional and when the input signal is a periodic oscillation. By the phase portrait analysis, we determine that the aperiodic oscillations are caused by periodically forced state transitions between a stable equilibrium and a stable limit cycle attractors around the saddle-node and saddle separatrix loop bifurcation points. It is revealed that the dynamical mechanism where the state crosses over the saddle-node and saddle separatrix loop bifurcations significantly contributes to the occurrence of chaotic oscillations forced by a periodic input. In addition, this mechanism can also give rise to chaotic oscillations in reciprocally connected Wilson-Cowan models. These results suggest that the dynamic attractor transition underlies chaotic behaviors in two coupled Wilson-Cowan oscillators. [ABSTRACT FROM AUTHOR]
- Published
- 2014
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42. Oscillations and Synchrony in a Network of Delayed Neural Masses
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Iain Pinder and Jonathan J. Crofts
- Subjects
Bifurcation theory ,Computational neuroscience ,Coupling (computer programming) ,Computer science ,Ordinary differential equation ,Network science ,Delay differential equation ,Complex network ,Topology ,Wilson–Cowan model - Abstract
In this chapter, we study the dynamics of a network of delayed Wilson–Cowan (WC) masses. We begin by analysing a single WC mass without delay, using numerical simulations and tools from numerical bifurcation theory to interrogate its dynamical behaviour under variation of important system parameters. We then briefly review the necessary prerequisites in network science before considering some of the ways in which delay differential equations differ from ordinary differential equations. Next we extend our analysis of the undelayed system to the case of a WC mass with two distinct discrete delays before reviewing some recent results from the literature. The chapter then moves on to study the effect that network structure has on the WC dynamics in the presence of both intra- and internodal delays. Deploying both artificial and experimental network structures, we find that network structure, coupled with delay times, can have both a regularising and deregularising effect, depending upon the overall strength of network coupling. Importantly, these results suggest that the presence of delays in a weakly coupled system provides a mechanism for avoiding undesirable (disordered) states, whilst still allowing the system to retain a sufficiently rich repertoire of network behaviours.
- Published
- 2020
43. Study on Induced Factors and Power Variations of the Steady-State Visual Evoked Potential Subharmonics
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Peng-Fei Sun, Yu-Qin Li, Yu-Rong Qin, Huan Gao, and Ni Chen
- Subjects
Physics ,Subharmonic ,Visual cortex ,medicine.anatomical_structure ,Acoustics ,medicine ,Stimulus frequency ,Noise intensity ,Evoked potential ,Stimulus (physiology) ,Wilson–Cowan model ,Brain–computer interface - Abstract
Harmonic and subharmonic components are contained in the spectrum of the steady-state visual evoked potential (SSVEP). These components are widely used in the brain-computer interface (BCI) technology and neuroscience research. Subharmonics cannot be easily induced, and the mechanism of the power variation of these components are ambiguous. In this paper, the induced factors and power variation of the SSVEP subharmonics are studied by a visual cortex network model simulation. The SSVEP subharmonics with orders higher than 1/2 are easily suppressed by noise. Therefore, the power variation of the SSVEP subharmonics with 1/2 order, and their induction factors are further analyzed. Simulation results show that the stimulus frequency, stimulus intensity, and noise intensity are three important parameters affecting the generation and power variations of the SSVEP subharmonics. If the stimulus intensity is weak, the 1/2 subharmonics cannot be induced. When the stimulus intensity increases to a certain value, 1/2 subharmonics appear in the SSVEP, and their power increases sharply with the increasing stimulus intensity. However, when the stimulus intensity exceeds a certain threshold, the 1/2 subharmonics remain constant. The bandwidth of the stimulus frequency, which can induce 1/2 subharmonics increases with increasing stimulus intensity under the same noise condition. Generation of 1/2 subharmonics and its power are suppressed by noise.
- Published
- 2020
44. Phase-dependence of response curves to deep brain stimulation and their relationship: from essential tremor patient data to a Wilson–Cowan model
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Benoit Duchet, Rafal Bogacz, Gihan Weerasinghe, Christian Bick, Hayriye Cagnan, and Peter Brown
- Subjects
Deep brain stimulation ,Computer science ,medicine.medical_treatment ,Thalamus ,Neuroscience (miscellaneous) ,Stimulation ,lcsh:RC321-571 ,Amplitude response curve ,Phase dependence ,03 medical and health sciences ,0302 clinical medicine ,medicine ,lcsh:Neurosciences. Biological psychiatry. Neuropsychiatry ,030304 developmental biology ,Phase response curve ,0303 health sciences ,Essential tremor ,lcsh:Mathematics ,Research ,Patient data ,Focus model ,lcsh:QA1-939 ,medicine.disease ,Wilson–Cowan model ,Phase-locked stimulation ,Wilson Cowan model ,Neuroscience ,030217 neurology & neurosurgery - Abstract
Essential tremor manifests predominantly as a tremor of the upper limbs. One therapy option is high-frequency deep brain stimulation, which continuously delivers electrical stimulation to the ventral intermediate nucleus of the thalamus at about 130 Hz. Constant stimulation can lead to side effects, it is therefore desirable to find ways to stimulate less while maintaining clinical efficacy. One strategy, phase-locked deep brain stimulation, consists of stimulating according to the phase of the tremor. To advance methods to optimise deep brain stimulation while providing insights into tremor circuits, we ask the question: can the effects of phase-locked stimulation be accounted for by a canonical Wilson–Cowan model? We first analyse patient data, and identify in half of the datasets significant dependence of the effects of stimulation on the phase at which stimulation is provided. The full nonlinear Wilson–Cowan model is fitted to datasets identified as statistically significant, and we show that in each case the model can fit to the dynamics of patient tremor as well as to the phase response curve. The vast majority of top fits are stable foci. The model provides satisfactory prediction of how patient tremor will react to phase-locked stimulation by predicting patient amplitude response curves although they were not explicitly fitted. We also approximate response curves of the significant datasets by providing analytical results for the linearisation of a stable focus model, a simplification of the Wilson–Cowan model in the stable focus regime. We report that the nonlinear Wilson–Cowan model is able to describe response to stimulation more precisely than the linearisation.
- Published
- 2020
45. Wilson–Cowan Neuronal Interaction Models with Distributed Delays
- Author
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Anca Radulescu, Eva Kaslik, and Emanuel-Attila Kokovics
- Subjects
Quantitative Biology::Neurons and Cognition ,Discrete time and continuous time ,Kernel (image processing) ,Generalization ,Plane (geometry) ,Bounded function ,Applied mathematics ,Stability (probability) ,Wilson–Cowan model ,Domain (mathematical analysis) ,Mathematics - Abstract
A generalization of the well-known Wilson–Cowan model of excitatory and inhibitory interactions in localized neuronal populations is presented, by taking into consideration distributed time delays. A stability and bifurcation analysis is undertaken for the generalized model, with respect to two characteristic parameters of the system. The stability region in the characteristic parameter plane is determined and a comparison is given for several types of delay kernels. It is shown that if a weak Gamma delay kernel is considered, as in the original Wilson–Cowan model without time-coarse graining, the resulting stability domain is unbounded, while in the case of a discrete time delay, the stability domain is bounded. This fact reveals an essential difference between the two scenarios, reflecting the importance of a careful choice of delay kernels in the mathematical model. Numerical simulations are presented to substantiate the theoretical results. Important differences are also highlighted by comparing the generalized model with the original Wilson–Cowan model without time delays.
- Published
- 2020
46. Generalized activity equations for spiking neural network dynamics.
- Author
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Buice, Michael A. and Chow, Carson C.
- Subjects
NUMERICAL analysis ,ANALYTICAL mechanics ,FORCE & energy ,MECHANICS (Physics) - Abstract
Much progress has been made in uncovering the computational capabilities of spiking neural networks. However, spiking neurons will always be more expensive to simulate compared to rate neurons because of the inherent disparity in time scales-the spike duration time is much shorter than the inter-spike time, which is much shorter than any learning time scale. In numerical analysis, this is a classic stiff problem. Spiking neurons are also much more difficult to study analytically. One possible approach to making spiking networks more tractable is to augment mean field activity models with some information about spiking correlations. For example, such a generalized activity model could carry information about spiking rates and correlations between spikes self-consistently. Here, we will show how this can be accomplished by constructing a complete formal probabilistic description of the network and then expanding around a small parameter such as the inverse of the number of neurons in the network. The mean field theory of the system gives a rate-like description. The first order terms in the perturbation expansion keep track of covariances [ABSTRACT FROM AUTHOR]
- Published
- 2013
- Full Text
- View/download PDF
47. Scale-freeness or partial synchronization in neural mass phase oscillator networks
- Subjects
Criticality ,Wilson-Cowan model ,Phase dynamics ,Synchronization ,Freeman model ,Power laws - Abstract
Modeling and interpreting (partial) synchronous neural activity can be a challenge. We illustrate this by deriving the phase dynamics of two seminal neural mass models: the Wilson-Cowan firing rate model and the voltage-based Freeman model. We established that the phase dynamics of these models differed qualitatively due to an attractive coupling in the first and a repulsive coupling in the latter. Using empirical structural connectivity matrices, we determined that the two dynamics cover the functional connectivity observed in resting state activity. We further searched for two pivotal dynamical features that have been reported in many experimental studies: (1) a partial phase synchrony with a possibility of a transition towards either a desynchronized or a (fully) synchronized state; (2) long-term autocorrelations indicative of a scale-free temporal dynamics of phase synchronization. Only the Freeman phase model exhibited scale-free behavior. Its repulsive coupling, however, let the individual phases disperse and did not allow for a transition into a synchronized state. The Wilson-Cowan phase model, by contrast, could switch into a (partially) synchronized state, but it did not generate long-term correlations although being located close to the onset of synchronization, i.e. in its critical regime. That is, the phase-reduced models can display one of the two dynamical features, but not both.
- Published
- 2018
48. Non-normal amplification of stochastic quasicycles
- Author
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Niccolò Zagli, Sara Nicoletti, Timoteo Carletti, Duccio Fanelli, Roberto Livi, and Giacomo Innocenti
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FOS: Physical sciences ,Pattern Formation and Solitons (nlin.PS) ,01 natural sciences ,010305 fluids & plasmas ,symbols.namesake ,0103 physical sciences ,010306 general physics ,Quantitative Biology - Populations and Evolution ,stochastic dynamics, network, non normal matrices ,Condensed Matter - Statistical Mechanics ,Statistical and Nonlinear Physics ,Statistics and Probability ,Condensed Matter Physics ,Physics ,Coupling ,Hopf bifurcation ,Wilson-Cowan model ,Statistical Mechanics (cond-mat.stat-mech) ,Degree (graph theory) ,non-normality ,Mathematical analysis ,stochastic dynamics ,Populations and Evolution (q-bio.PE) ,Abscissa ,Disordered Systems and Neural Networks (cond-mat.dis-nn) ,Condensed Matter - Disordered Systems and Neural Networks ,Nonlinear Sciences - Pattern Formation and Solitons ,Wilson–Cowan model ,Loop (topology) ,Amplitude ,quasi cycle ,FOS: Biological sciences ,Phase space ,symbols - Abstract
Stochastic quasi-cycles for a two species model of the excitatory-inhibitory type, arranged on a triangular loop, are studied. By increasing the strength of the inter-nodes coupling, one moves the system towards the Hopf bifurcation and the amplitude of the stochastic oscillations are consequently magnified. When the system is instead constrained to evolve on specific manifolds, selected so as to return a constant rate of deterministic damping for the perturbations, the observed amplification correlates with the degree of non normal reactivity, here quantified by the numerical abscissa. The thermodynamics of the reactive loop is also investigated and the degree of inherent reactivity shown to facilitate the out-of-equilibrium exploration of the available phase space.
- Published
- 2018
49. Introducing divisive inhibition in the Wilson-Cowan model
- Author
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Yujiang Wang and Christoforos A. Papasavvas
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Formalism (philosophy of mathematics) ,Cortical circuits ,Quantitative Biology::Neurons and Cognition ,Computer science ,Inhibitory postsynaptic potential ,Neuroscience ,Wilson–Cowan model - Abstract
Both subtractive and divisive inhibition has been recorded in cortical circuits and recent findings suggest that different interneuronal populations are responsible for the different types of inhibition. This calls for the formulation and description of these inhibitory mechanisms in computational models of cortical networks. Neural mass and neural field models typically only feature subtractive inhibition. Here, we introduce how divisive inhibition can be incorporated in such models, using the Wilson-Cowan modelling formalism as an example. In addition, we show how the subtractive and divisive modulations can be combined. Including divisive inhibition in neural mass models is a crucial step in understanding its role in shaping oscillatory phenomena in cortical networks.
- Published
- 2019
- Full Text
- View/download PDF
50. Traveling waves in a spatially-distributed Wilson–Cowan model of cortex: From fronts to pulses
- Author
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G. Bard Ermentrout and Jeremy D. Harris
- Subjects
0301 basic medicine ,Physics ,education.field_of_study ,Quantitative Biology::Neurons and Cognition ,Wave propagation ,Population ,Front (oceanography) ,Spatiotemporal pattern ,Statistical and Nonlinear Physics ,Condensed Matter Physics ,Instability ,Stable manifold ,Wilson–Cowan model ,Pulse (physics) ,03 medical and health sciences ,030104 developmental biology ,0302 clinical medicine ,Classical mechanics ,education ,030217 neurology & neurosurgery - Abstract
Wave propagation in excitable media has been studied in various biological, chemical, and physical systems. Waves are among the most common evoked and spontaneous organized activity seen in cortical networks. In this paper, we study traveling fronts and pulses in a spatially-extended version of the Wilson–Cowan equations, a neural firing rate model of sensory cortex having two population types: Excitatory and inhibitory. We are primarily interested in the case when the local or space-clamped dynamics has three fixed points: (1) a stable down state; (2) a saddle point with stable manifold that acts as a threshold for firing; (3) an up state having stability that depends on the time scale of the inhibition. In the case when the up state is stable, we look for wave fronts, which transition the media from a down to up state, and when the up state is unstable, we are interested in pulses, a transient increase in firing that returns to the down state. We explore the behavior of these waves as the time and space scales of the inhibitory population vary. Some interesting findings include bistability between a traveling front and pulse, fronts that join the down state to an oscillation or spatiotemporal pattern, and pulses which go through an oscillatory instability.
- Published
- 2018
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