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The mechanisms responsible for pattern formation have attracted a great deal of attention since Alan Turing elucidated his fascinating idea on diffusion-induced instability of steady states. Subsequent studies on the models demonstrated an entirely different class of solutions; namely localized structures composing of steadily moving fronts and pulses. In such energy-driven motion, the combination of short and long-range interaction plays an important ingredient for the generation of complex patterns. This competition on traveling wave dynamics, commonly observed in many physical and chemical phenomena, will be highlighted. [ABSTRACT FROM AUTHOR]

Understanding how genetically encoded rules drive and guide complex neuronal growth processes is essential to comprehending the brain’s architecture, and agent-based models (ABMs) offer a powerful simulation approach to further develop this understanding. However, accurately calibrating these models remains a challenge. Here, we present a novel application of Approximate Bayesian Computation (ABC) to address this issue. ABMs are based on parametrized stochastic rules that describe the time evolution of small components–the so-called agents–discretizing the system, leading to stochastic simulations that require appropriate treatment. Mathematically, the calibration defines a stochastic inverse problem. We propose to address it in a Bayesian setting using ABC. We facilitate the repeated comparison between data and simulations by quantifying the morphological information of single neurons with so-called morphometrics and resort to statistical distances to measure discrepancies between populations thereof. We conduct experiments on synthetic as well as experimental data. We find that ABC utilizing Sequential Monte Carlo sampling and the Wasserstein distance finds accurate posterior parameter distributions for representative ABMs. We further demonstrate that these ABMs capture specific features of pyramidal cells of the hippocampus (CA1). Overall, this work establishes a robust framework for calibrating agent-based neuronal growth models and opens the door for future investigations using Bayesian techniques for model building, verification, and adequacy assessment. [ABSTRACT FROM AUTHOR]

Calcium is a decisive messenger for neuronal vivid functions. The calcium intracellular sequestering major unit is the Endoplasmic Reticulum (ER). Brownian motion of calcium could be bound to different buffers like S100B, calmodulin, etc, and different organelles. Plasma membrane channels like voltage-gated calcium channels (VGCC) and Plasma Membrane Calcium ATPase (PMCA), Orai channel could perturb the calcium concentration. To investigate the calcium interplay for intracellular signaling we have developed the two-dimensional time fractional reaction–diffusion equation. To solve this model analytically, we have used the Laplace and Fourier cosine integral transform method. By using Green's function we obtained the compact solution in closed form with Mainardi's function and Wright's function. Uniqueness and existence proved the more fundamental approach to the fractional reaction–diffusion problem. The fractional Caputo approach gives better insight into this real-life problem by its nonlocal nature. Significant effects of different parameters on free calcium ions were obtained and the results are interpreted with normal and Alzheimeric cells. Non-local property and dynamical aspects are graphically presented which might provide insight into the Stromal interaction molecule (STIM) and S100B parameters. [ABSTRACT FROM AUTHOR]

To understand the dynamics of neurodegenerative disease, we consider a fractional-order yeast prion mathematical model. We characterize the system dynamics by locating the steady states and perform their local stability analysis both theoretically and numerically. We employ the homotopy perturbation method to determine approximate solutions for the proposed model. By varying the fractional orders, we generate plots of these solutions over time to examine the dynamics of the Sup35 monomer and Sup35 prion populations. Our simulations reveal that the dynamical behaviors of both populations can undergo alterations depending upon the chosen fractional orders and the value of the threshold parameter. [ABSTRACT FROM AUTHOR]

Dynamical systems on networks typically involve several dynamical processes evolving at different timescales. For instance, in Alzheimer’s disease, the spread of toxic protein throughout the brain not only disrupts neuronal activity but is also influenced by neuronal activity itself, establishing a feedback loop between the fast neuronal activity and the slow protein spreading. Motivated by the case of Alzheimer’s disease, we study the multiple-timescale dynamics of a heterodimer spreading process on an adaptive network of Kuramoto oscillators. Using a minimal two-node model, we establish that heterogeneous oscillatory activity facilitates toxic outbreaks and induces symmetry breaking in the spreading patterns. We then extend the model formulation to larger networks and perform numerical simulations of the slow-fast dynamics on common network motifs and on the brain connectome. The simulations corroborate the findings from the minimal model, underscoring the significance of multiple-timescale dynamics in the modeling of neurodegenerative diseases. [ABSTRACT FROM AUTHOR]

First-principles-based modelings have been extremely successful in providing crucial insights and predictions for complex biological functions and phenomena. However, they can be hard to build and expensive to simulate for complex living systems. On the other hand, modern data-driven methods thrive at modeling many types of high-dimensional and noisy data. Still, the training and interpretation of these data-driven models remain challenging. Here, we combine the two types of methods to model stochastic neuronal network oscillations. Specifically, we develop a class of artificial neural networks to provide faithful surrogates to the high-dimensional, nonlinear oscillatory dynamics produced by a spiking neuronal network model. Furthermore, when the training data set is enlarged within a range of parameter choices, the artificial neural networks become generalizable to these parameters, covering cases in distinctly different dynamical regimes. In all, our work opens a new avenue for modeling complex neuronal network dynamics with artificial neural networks. [ABSTRACT FROM AUTHOR]

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]

In this paper, we study monotone properties of random and stochastic functional differential equations and their global dynamics. First, we show that random functional differential equations (RFDEs) generate the random dynamical system (RDS) if and only if all the solutions are globally defined, and establish the comparison theorem for RFDEs and the random Riesz representation theorem. These three results lead to the Borel measurability of coefficient functions in the Riesz representation of variational equations for quasimonotone RFDEs, which paves the way following the Smith line to establish eventual strong monotonicity for the RDS under cooperative and irreducible conditions. Then strong comparison principles, strong sublinearity theorems and the existence of random attractors for RFDEs are proved. Finally, criteria are presented for the existence of a unique random equilibrium and its global stability in the universe of all the tempered random closed sets of the positive cone. Applications to typical random or stochastic delay models in monotone dynamical systems, such as biochemical control circuits, cyclic gene models and Hopfield-type neural networks, are given. [ABSTRACT FROM AUTHOR]

This article presents a biological neural network model driven by inhomogeneous Poisson processes accounting for the intrinsic randomness of synapses. The main novelty is the introduction of sparse interactions: each firing neuron triggers an instantaneous increase in electric potential to a fixed number of randomly chosen neurons. We prove that, as the number of neurons approaches infinity, the finite network converges to a nonlinear mean-field process characterised by a jump-type stochastic differential equation. We show that this process displays a phase transition: the activity of a typical neuron in the infinite network either rapidly dies out, or persists forever, depending on the global parameters describing the intensity of interconnection. This provides a way to understand the emergence of persistent activity triggered by weak input signals in large neural networks. [ABSTRACT FROM AUTHOR]

This work is devoted to study a new class of static neural network. We propose a static multidirectional associative memory neural network with time-varying coefficients and delays. We assess the existence and global exponential stability of weighted pseudo almost-periodic solutions for the systems through exponential dichotomy theory, contraction mapping fixed point theorem, differential inequality techniques and weighted pseudo almost-periodic functions. The theoretical results obtained are supported with numerical simulations. To our knowledge, no paper in the literature has investigated the existence and the delay-independent stability of weighted pseudo almost-periodic solution for static delayed MAM neural network. The results obtained are new and complements that found in the literature. [ABSTRACT FROM AUTHOR]

We investigate the Cohen-Grosberg differential equations with mixed delays and time-varying coefficient: Several useful results on the functional space of such functions like completeness and composition theorems are established. By using the fixed-point theorem and some properties of the doubly measure pseudo almost automorphic functions, a set of sufficient criteria are established to ensure the existence, uniqueness and global exponential stability of a (μ, ν)-pseudo almost automorphic solution. The theory of this work generalizes the classical results on weighted pseudo almost automorphic functions. Finally, a numerical example is provided to illustrate the validity of the proposed theoretical results. [ABSTRACT FROM AUTHOR]

In this review, we are considering an actively developing tool in neuroscience—robotic modeling. The new perspective and existing application fields, tools, and methods are discussed. We try to determine starting positions and approaches that are useful at the beginning of new research in this field. Among multiple directions of the research is robotic modeling on the level of muscles fibers and their afferents, skin surface sensors, muscles, and joints proprioceptors. Some examples of technical implementation for physical modeling are reviewed. They are software and hardware tools like event-related modeling algorithms, reduced neuron models, robotic drives constructions. We observe existing drives technologies and prospective electric motor types: switched reluctance and transverse flux motors. Next, we look at the existing examples and approaches for robotic modeling of the cerebellum and spinal cord neural networks. These examples show practical methods for the model neural network architecture and adaptation. Those methods allow the use of cortical and spinal cord reflexes for the network training and apply additional artificial blocks for data processing in other brain structures that transmit and receive data from biologically realistic models. [ABSTRACT FROM AUTHOR]

A dynamical system with a plastic self-organising velocity vector field was introduced in Janson and Marsden (Sci Rep 7:17007, 2017) as a mathematical prototype of new explainable intelligent systems. Although inspired by the brain plasticity, it does not model or explain any specific brain mechanisms or processes, but instead expresses a hypothesised principle possibly implemented by the brain. The hypothesis states that, by means of its plastic architecture, the brain creates a plastic self-organising velocity vector field, which embodies self-organising rules governing neural activity and through that the behaviour of the whole body. The model is represented by a two-tier dynamical system, in which the observable behaviour obeys a velocity field, which is itself controlled by another dynamical system. Contrary to standard brain models, in the new model the sensory input affects the velocity field directly, rather than indirectly via neural activity. However, this model was postulated without sufficient explication or theoretical proof of its mathematical consistency. Here we provide a more rigorous mathematical formulation of this problem, make several simplifying assumptions about the form of the model and of the applied stimulus, and perform its mathematical analysis. Namely, we explore the existence, uniqueness, continuity and smoothness of both the plastic velocity vector field controlling the observable behaviour of the system, and the of the behaviour itself. We also analyse the existence of pullback attractors and of forward limit sets in such a non-autonomous system of a special form. Our results verify the consistency of the problem and pave the way to constructing more models with specific pre-defined cognitive functions. [ABSTRACT FROM AUTHOR]

A new mathematical model for complex neural networks of the partly diffusive Hindmasrh-Rose equations with boundary coupling is proposed. Through analysis of absorbing dynamics for the solution semiflow, the asymptotic synchronization of the complex neuronal networks at a uniform exponential rate is proved under the condition that stimulation signal strength of the ensemble boundary coupling exceeds a quantitative threshold expressed by the biological parameters. [ABSTRACT FROM AUTHOR]

The establishment of a functioning neuronal network is a crucial step in neural development. During this process, neurons extend neurites—axons and dendrites—to meet other neurons and interconnect. Therefore, these neurites need to migrate, grow, branch and find the correct path to their target by processing sensory cues from their environment. These processes rely on many coupled biophysical effects including elasticity, viscosity, growth, active forces, chemical signaling, adhesion and cellular transport. Mathematical models offer a direct way to test hypotheses and understand the underlying mechanisms responsible for neuron development. Here, we critically review the main models of neurite growth and morphogenesis from a mathematical viewpoint. We present different models for growth, guidance and morphogenesis, with a particular emphasis on mechanics and mechanisms, and on simple mathematical models that can be partially treated analytically. [ABSTRACT FROM AUTHOR]

We propose a novel differential equation model assuming the stage theory (trichromatic theory and opponent-process theory) for color inputs based on our previous model for light-dark inputs, which is an extension of the lateral inhibition model proposed by Peskin (Partial Differential Equations in Biology: Courant Institute of Mathematical Sciences Lecture Notes, New York, 1976). First, we consider the stationary problem in our novel model for color inputs and show that the output of our model can be described by a convolution integral. Moreover, we derive the necessary and sufficient conditions for the appearance of Mexican hat-type integral kernels in the outputs of our model for color inputs. Second, we demonstrate numerical results for simple color inputs and provide a theoretical prediction that the self-control mechanism exerted at horizontal cells, in conjunction with the opponent-colors (red-green, yellow-blue, and light-dark (or white-black)) plays an important role for the occurrence and non-occurrence of typical color contrasts. [ABSTRACT FROM AUTHOR]

Computational modelling of biochemical reaction pathways is an increasingly important part of neuroscience research. In order to be useful, computational models need to be valid in two senses: First, they need to be consistent with experimental data and able to make testable predictions (external validity). Second, they need to be internally consistent and independently reproducible (internal validity). Here, we discuss both types of validity and provide a brief overview of tools and technologies used to ensure they are met. We also suggest the introduction of new collaborative technologies to ensure model validity: an incentivised experimental database for external validity and reproducibility audits for internal validity. Both rely on FAIR principles and on collaborative science practices. [ABSTRACT FROM AUTHOR]

Traveling phase waves are commonly observed in recordings of the cerebral cortex and are believed to organize behavior across different areas of the brain. We use this as motivation to analyze a one-dimensional network of phase oscillators that are nonlocally coupled via the phase response curve (PRC) and the Dirac delta function. Existence of waves is proven and the dispersion relation is computed. Using the theory of distributions enables us to write and solve an associated stability problem. First and second order perturbation theory is applied to get analytic insight and we show that long waves are stable while short waves are unstable. We apply the results to PRCs that come from mitral neurons. We extend the results to smooth pulse-like coupling by reducing the nonlocal equation to a local one and solving the associated boundary value problem. [ABSTRACT FROM AUTHOR]

Acetylcholine (ACh), one of the brain's most potent neuromodulators, can affect intrinsic neuron properties through blockade of an M-type potassium current. The effect of ACh on excitatory and inhibitory cells with this potassium channel modulates their membrane excitability, which in turn affects their tendency to synchronize in networks. Here, we study the resulting changes in dynamics in networks with inter-connected excitatory and inhibitory populations (E–I networks), which are ubiquitous in the brain. Utilizing biophysical models of E–I networks, we analyze how the network connectivity structure in terms of synaptic connectivity alters the influence of ACh on the generation of synchronous excitatory bursting. We investigate networks containing all combinations of excitatory and inhibitory cells with high (Type I properties) or low (Type II properties) modulatory tone. To vary network connectivity structure, we focus on the effects of the strengths of inter-connections between excitatory and inhibitory cells (E–I synapses and I–E synapses), and the strengths of intra-connections among excitatory cells (E–E synapses) and among inhibitory cells (I-I synapses). We show that the presence of ACh may or may not affect the generation of network synchrony depending on the network connectivity. Specifically, strong network inter-connectivity induces synchronous excitatory bursting regardless of the cellular propensity for synchronization, which aligns with predictions of the PING model. However, when a network's intra-connectivity dominates its inter-connectivity, the propensity for synchrony of either inhibitory or excitatory cells can determine the generation of network-wide bursting. [ABSTRACT FROM AUTHOR]

This paper investigates the fixed-time synchronization problems for competitive neural networks with proportional delays and impulsive effect. The concerned network involves two coupling terms, i.e., long-term memory and short-term memory, which leads to the difficulty to the dynamics analysis. Based on Lyapunov functionals, the differential inequalities and for the objective of making the settling time independent of initial condition, a novel criterion guaranteeing the fixed-time synchronization of addressed system is derived. Finally, two examples and their simulations are given to demonstrate the effectiveness of the obtained results. [ABSTRACT FROM AUTHOR]

By means of the fixed-point methods and the properties of the μ-pseudo almost periodic functions, we prove the existence, uniqueness, and exponential stability of the μ-pseudo almost periodic solutions for some models of recurrent neural networks with mixed delays and time-varying coefficients, where μ is a positive measure. A numerical example is given to illustrate our main results. [ABSTRACT FROM AUTHOR]

This paper is concerned with a hematopoiesis model with mixed delays. Under new conditions, we studied the existence, uniqueness and global exponential stability of pseudo almost automorphic solutions for the suggested model. Our approach is mainly based on the exponential dichotomy of linear differential equation, Banach's fixed-point principle and suitable Lyapunov functional. At the end, some numerical examples are presented to demonstrate the effectiveness of our findings. [ABSTRACT FROM AUTHOR]

This article examines the existence of the unique piecewise pseudo almost periodic for impulsive fuzzy cellular neural networks by using the contraction mapping principle and piecewise pseudo almost periodic function theory. Further, sufficient certain conditions for their global exponential stability are produced through the use of differential inequality and generalized Gronwall–Bellman inequality. Our results are new and complement some previously known ones. Two examples and their numerical simulations are performed to ensure our theoretical results. [ABSTRACT FROM AUTHOR]

This article is concerned with a high-order Hopfield bidirectional associative memory neural networks with time-varying coefficients and mixed delays. Sufficient conditions are derived for the existence, the uniqueness and the exponential stability of (μ , ν) -pseudo-almost automorphic solutions of the considered model. Banach fixed-point theorem is applied for the existence and the uniqueness results. Global exponential stability is derived via differential inequalities. Finally, two examples are provided to support the feasibility of the theoretical results. [ABSTRACT FROM AUTHOR]

The effect of gap junctions as well as the biological mechanisms behind seizure wave propagation is not completely understood. In this work, we use a simple neural field model to study the possible influence of gap junctions specifically on cortical wave propagation that has been observed in vivo preceding seizure termination. We consider a voltage-based neural field model consisting of an excitatory and an inhibitory population as well as both chemical and gap junction-like synapses. We are able to approximate important properties of cortical wave propagation previously observed in vivo before seizure termination. This model adds support to existing evidence from models and clinical data suggesting a key role of gap junctions in seizure wave propagation. In particular, we found that in this model gap junction-like connectivity determines the propagation of one-bump or two-bump traveling wave solutions with features consistent with the clinical data. For sufficiently increased gap junction connectivity, wave solutions cease to exist. Moreover, gap junction connectivity needs to be sufficiently low or moderate to permit the existence of linearly stable solutions of interest. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying the finite-time and fixed-time of inertial Cohen–Grossberg type neural networks (ICGNNs) with time varying delays. First, by constructing a proper variable substitution, the original (ICGNNs) can be rewritten as first-order differential system. Second, by utilizing feedback controllers and constructing suitable Lyapunov functionals, several new sufficient conditions guaranteeing the finite-time and the fixed-time synchronization of ICGNNs with time varying delays are obtained based on different finite-time synchronization analysis techniques. The obtained sufficient conditions are simple and easy to verify. Numerical simulations are given to illustrate the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

This paper is concerned with an impulsive non-autonomous high-order Hopfield neural network with mixed delays. Under proper conditions, we studied the existence, the uniqueness and the global exponential stability of asymptotic almost automorphic solutions for the suggested system. Our method was mainly based on the Banach's fixed-point theorem and the generalized Gronwall–Bellman inequality. Moreover, four examples are presented to demonstrate the effectiveness of the proposed findings. [ABSTRACT FROM AUTHOR]

We study stochastic Amari-type neural field equations, which are mean-field models for neural activity in the cortex. We prove that under certain assumptions on the coupling kernel, the neural field model can be viewed as a gradient flow in a nonlocal Hilbert space. This makes all gradient flow methods available for the analysis, which could previously not be used, as it was not known, whether a rigorous gradient flow formulation exists. We show that the equation is well-posed in the nonlocal Hilbert space in the sense that solutions starting in this space also remain in it for all times and space-time regularity results hold for the case of spatially correlated noise. Uniqueness of invariant measures, ergodic properties for the associated Feller semigroups, and several examples of kernels are also discussed. [ABSTRACT FROM AUTHOR]

This paper deals with a generalized multidirectional associative memory neural network with time-varying coefficients and mixed delays. Some criteria are established for the existence and the global exponential stability of weighted pseudo almost-periodic solutions for the studied model. To do so, the exponential dichotomy theory, the contraction mapping fixed point theorem and the differential inequality techniques are used. Moreover, an example and its numerical simulation are employed to illustrate the main results. [ABSTRACT FROM AUTHOR]

This paper is concerned with a class of neutral type recurrent neural networks with time-varying delays, distributed delay and D operator on time–space scales which unify the continuous-time and the discrete-time recurrent neural networks under the same framework. Some sufficient conditions are given for the existence and the global exponential stability of the pseudo almost periodic solution by using inequality analysis techniques on time scales, fixed point theorem and the theory of calculus on time scales. An example is given to show the effectiveness of the derived results via computer simulations. [ABSTRACT FROM AUTHOR]

In this paper, by employing contraction mapping principle theorem and by finding suitable Lyapunov–Krasovskii functional, some sufficient conditions are given for the existence and the global exponential stability, the uniform asymptotic stability, the global asymptotic stability and the uniform stability of the unique equilibrium point of bidirectional associative memory neural networks with distributed delays and leakage time-varying delays. Three illustrative examples are given at the end of this paper to demonstrate the effectiveness of our theoretical results. [ABSTRACT FROM AUTHOR]

In the paper, under some sufficient conditions, the existence and the global exponential stability of a pseudo almost automorphic solution to multidirectional associative memory (MAM) neural network with time-varying coefficients and mixed delays is obtained by combining both differential inequality techniques and the classical Banach's fixed-point principle. As you will see, our results improve and generalize many previous known results in MAM. Finally, an example together with its numeric simulations show the feasibility and effectiveness of our main results. [ABSTRACT FROM AUTHOR]

This paper is concerned with an impulsive Cohen–Grossberg neural networks with mixed delays. Under proper conditions, we studied the existence, the uniqueness and the global exponential stability of asymptotic almost automorphic solutions for the suggested system. Our method was mainly based on the Banach's fixed point theorem, the generalized Gronwall–Bellman inequality and the Lyapunov functional method. Moreover, two examples are presented to demonstrate the effectiveness of the proposed findings. [ABSTRACT FROM AUTHOR]

We study whether the transition from wakefulness to light sleep is a synergetic phase transition in the brain. [ABSTRACT FROM AUTHOR]

Recently, several experiments have demonstrated the existence of fractional diffusion in the neuronal transmission occurring in the Purkinje cells, whose malfunctioning is known to be related to the lack of voluntary coordination and the appearance of tremors. Also, a classical mathematical feature is that (fractional) parabolic equations possess smoothing effects, in contrast with the case of hyperbolic equations, which typically exhibit shocks and discontinuities. In this paper, we show how a simple toy-model of a highly ramified structure, somehow inspired by that of the Purkinje cells, may produce a fractional diffusion via the superposition of travelling waves that solve a hyperbolic equation. This could suggest that the high ramification of the Purkinje cells might have provided an evolutionary advantage of “smoothing” the transmission of signals and avoiding shock propagations (at the price of slowing a bit such transmission). Although an experimental confirmation of the possibility of such evolutionary advantage goes well beyond the goals of this paper, we think that it is intriguing, as a mathematical counterpart, to consider the time fractional diffusion as arising from the superposition of delayed travelling waves in highly ramified transmission media. The case of a travelling concave parabola with sufficiently small curvature is explicitly computed. The new link that we propose between time fractional diffusion and hyperbolic equation also provides a novelty with respect to the usual paradigm relating time fractional diffusion with parabolic equations in the limit. This paper is written in such a way as to be of interest to both biologists and mathematician alike. In order to accomplish this aim, both complete explanations of the objects considered and detailed lists of references are provided. [ABSTRACT FROM AUTHOR]

We introduce an asymptotic expansion of the transition density of a nonlinear oscillator involving a jump-diffusion process. We approximate the transition density by expanding the characteristic function of the solution to the nonlinear oscillator with respect to time and present a numerical verification of the asymptotic expansion of the transition density. This study provides us with a new mathematical framework for analyzing the dynamics of stochastic mathematical neuronal models using a jump-diffusion process. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the exact simulation of a class of piecewise deterministic Markov processes. We show how to perform an efficient thinning algorithm depending on the jump rate bound. For different types of bounds, we compare theoretically the efficiency of the algorithm (measured by the mean ratio between the total number of jump times generated by thinning and the number of selected ones) and we compare numerically the computation times. We use the thinning algorithm on Hodgkin-Huxley models with Markovian ion channels dynamics to illustrate our results. [ABSTRACT FROM AUTHOR]

This paper proposes a differential equation model of the human vision system. Our model has a hierarchical structure in the known retinal cell neural network, and hence, enable to explain what aspects of behaviors of the vision system is affected by which parameters. As the result, our model possesses the following two characteristics: First, it enable the derivation of an integral equation model with a Mexican-hat shape kernel as a necessary result of mechanisms (reduction of the retinal image resolution and a self-control mechanism formed by non-local interaction), in contrast to previous models that assumed various types of Mexican-hat shapes a priori. Second, it can explain two mutually contradicting phenomena called lightness contrast and assimilation. Moreover, our model explains the reason why lightness optical illusions do or do not occur via the magnitude of a control parameter. [ABSTRACT FROM AUTHOR]

We use conductance based neuron models, and the mathematical modeling of optogenetics to define controlled neuron models and we address the minimal time control of these affine systems for the first spike from equilibrium. We apply tools of geometric optimal control theory to study singular extremals, and we implement a direct method to compute optimal controls. When the system is too large to theoretically investigate the existence of singular optimal controls, we observe numerically the optimal bang-bang controls. [ABSTRACT FROM AUTHOR]

Diffuse brain injury is caused by rapid rotation of the head, and causes strain injury to tissue throughout the brain. Following strain injury, axons exhibit delayed recovery, showing regional buckling behavior immediately after stretch and returning to their original appearance over an extended period of time. This axonal buckling is hypothesized to occur as a result of localized stretching within the axon: Rapid strain causes mechanical damage to microtubules, increasing the effective length of axons. This damage is repaired gradually returning the axon to its initial length. Here, we test the hypothesis that localized stretching is a possible explanation for the regional buckling behavior. An elongated region of axon is modeled as an Euler beam on an elastic foundation, where the foundation represents the surrounding brain tissue, which consists of glial cells and extracellular matrix. After stretch the elastic foundation returns immediately to its pre-stretch length, while the axon is initially elongated and returns to its original length over a longer period of time. The model exhibits solutions similar to those observed experimentally in post-stretch axons, with undulations that have a similar wavelength and amplitude. [ABSTRACT FROM AUTHOR]

In this paper, we shall explain a new result concerning piecewise weighted pseudo almost-periodic solution of impulsive recurrent neural networks with time-varying coefficients and mixed delays. Precisely, some sufficient conditions are given to prove the existence and the exponential stability of piecewise weighted pseudo almost-periodic solution by employing fixed point theorem, generalized Gronwall-Bellman inequality and differential inequality techniques. Finally, an illustrative example is given to demonstrate the effectiveness of our results. [ABSTRACT FROM AUTHOR]

Four classical examples on mathematical modeling in the life-sciences are summarized. These include Turing's diffusion-reaction systems in morphogenesis, Hodkin's and Huxley's model on the initiation and propagation of action potentials in a nerve fibre, first rigorous chemotaxis models, and the mathematical analysis of molecular sequence characteristics. [ABSTRACT FROM AUTHOR]

Many neuronal circuits driving coordinated locomotion are composed of chains of half-center oscillators (HCOs) of various lengths. The HCO is a common motif in central pattern generating circuits (CPGs); an HCO consists of two neurons, or two neuronal populations, connected by reciprocal inhibition. To maintain appropriate motor coordination for effective locomotion over a broad range of frequencies, chains of CPGs must produce approximately constant phase-differences in a robust manner. In this article, we study phase-locking in chains of nearest-neighbor coupled HCOs and examine how the circuit architecture can promote phase-constancy, i.e., inter-HCO phase-differences that are frequency-invariant. We use two models with different levels of abstraction: (1) a conductance-based model in which each neuron is modeled by the Morris-Lecar equations (the ML-HCO model); and (2) a coupled phase model in which the state of each HCO is captured by its phase (the phase-HCO model). We show that one of four phase-waves with inter-HCO phase-differences at approximately 0, 25, 50 or 75 % arises robustly as a result of the inter-HCO connection topology, and its robust existence is not affected by the number of HCOs in the chain, the difference in strength between the ascending and descending nearest-neighbor connections, or the number of nearest-neighbor connections. Our results show that the internal anti-phase structure of the HCO and an appropriate inter-HCO connection topology together can provide a mechanism for robust (i.e., frequency-independent) limb coordination in segmented animals, such as the 50 % interlimb phase-differences in the tripod gate of stick insects and cockroaches, and the 25 % interlimb phase-differences in crayfish and other long-tailed crustaceans during forward swimming. [ABSTRACT FROM AUTHOR]

Existence of piecewise differentiable pseudo almost-periodic solutions for a class of impulsive high-order Hopfield neural networks with leakage delays are established by employing the fixed point theorem, differential inequality and Lyapunov functionals. The results of this paper are new and they supplement previously known works. Numerical example with graphical illustration is given to illuminate our main results. [ABSTRACT FROM AUTHOR]

Burst suppression, a pattern of the electroencephalogram characterized by quasi-periodic alternation of high-voltage activity (burst) and isoelectric silence (suppression), is typically associated with states of unconsciousness, such as in deep general anesthesia and certain etiologies of coma. Recent computational models for burst suppression have attributed the slow (up to tens of seconds) time-scale of burst termination and re-initiation to cycling in supportive physiological process, such as cerebral metabolism. That is, activity-dependent substrate ('energy') depletion during bursts, followed by substrate recovery during suppression. Such a model falls into the category of a fast-slow dynamical system, commonly used to describe neuronal bursting more generally. Here, following this basic paradigm, we develop a low dimensional mean field model for burst suppression that adds several new features and capabilities to previous models. Most notably, this new model includes explicit homeostatic interactions wherein the rates of substrate recovery are tied to neuronal activity in a supply demand loop, creating a physiologically consistent, reciprocal interaction between the neural and substrate processes. We develop formal analysis of the model dynamics, showing, in particular, the capability of the model to produce burst-like activity as a consequence of neuronal downregulation only, without any direct perturbation to the substrate dynamics. Further, we use a synchronization analysis to contrast different mechanisms for spatially local versus global bursting. The analysis performed generates characterizations that are consistent with experimental observations of spatiotemporal features such as burst onset, duration, and spatial organization and, moreover, generates predictions regarding the presence of bistability and hysteresis in the underlying system. Thus, the model provides new dynamical insight into the mechanisms of burst suppression and, moreover, a tractable platform for more detailed future characterizations. [ABSTRACT FROM AUTHOR]

Homeostatic processes that provide negative feedback to regulate neuronal firing rates are essential for normal brain function. Indeed, multiple parameters of individual neurons, including the scale of afferent synapse strengths and the densities of specific ion channels, have been observed to change on homeostatic time scales to oppose the effects of chronic changes in synaptic input. This raises the question of whether these processes are controlled by a single slow feedback variable or multiple slow variables. A single homeostatic process providing negative feedback to a neuron's firing rate naturally maintains a stable homeostatic equilibrium with a characteristic mean firing rate; but the conditions under which multiple slow feedbacks produce a stable homeostatic equilibrium have not yet been explored. Here we study a highly general model of homeostatic firing rate control in which two slow variables provide negative feedback to drive a firing rate toward two different target rates. Using dynamical systems techniques, we show that such a control system can be used to stably maintain a neuron's characteristic firing rate mean and variance in the face of perturbations, and we derive conditions under which this happens. We also derive expressions that clarify the relationship between the homeostatic firing rate targets and the resulting stable firing rate mean and variance. We provide specific examples of neuronal systems that can be effectively regulated by dual homeostasis. One of these examples is a recurrent excitatory network, which a dual feedback system can robustly tune to serve as an integrator. [ABSTRACT FROM AUTHOR]

In this article, we consider a class of neutral impulsive shunting inhibitory cellular neural networks with time varying coefficients and leakage delays. We study the existence and the exponential stability of the piecewise differentiable pseudo almost-periodic solutions and establish sufficient conditions for the existence and exponential stability of such solutions. An example is provided to illustrate the theory developed in this work. [ABSTRACT FROM AUTHOR]

Slow oscillations in firing rate of neural populations are commonly observed during slow wave sleep. These oscillations are partitioned into up and down states, where the population switches between high and low firing rates (Sanchez-Vives and McCormick in Nat Neurosci 3:1027-1034, 2000). Transitions between up and down states can be synchronized at considerably long ranges (Volgushev et al. in J Neurosci 26:5665-5672, 2006). To explore how external perturbations shape the phase of slow oscillations, we analyze a reduced model of up and down state transitions involving a population neural activity variable and a global adaptation variable. The adaptation variable represents the average of all the slow hyperpolarizing currents received by neurons in a large population. Recurrent connectivity leads to a bistable neural population, where a low firing rate state coexists with a high firing rate state, where persistent activity is maintained via excitatory connections. Adaptation eventually inactivates the high activity state, and the low activity state then persists until adaptation has significantly decayed. We analyze the phase response of the rate model by taking advantage of the separation of timescales between the fast activity and slow adaptation variables. This analysis reveals that perturbations to the neural activity variable have a considerably weaker effect on the oscillation phase than adaptation perturbations. When noise is not incorporated into the rate model, the period of the slow oscillation is determined by the timescale of the slow adaptation variable. In the presence of noise, times at which the population transitions between the low and high activity states become variable. This is because the rise and decay of the adaptation variable is now stochastically-driven, leading to a distribution of transition times. Interestingly, common noise in the adaptation variable can lead to a correlation of two distinct slow oscillating populations. This effect is still significant in the event that each population contains its own local sources of noise. We also show this phenomenon can occur a spiking network. Our results demonstrate the relative contributions of excitatory input and hyperpolarizing current fluctuations on the phase of slow oscillations. [ABSTRACT FROM AUTHOR]