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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]

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]

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]

In volume transmission (or neuromodulation) neurons do not make one-to-one connections to other neurons, but instead simply release neurotransmitter into the extracellular space from numerous varicosities. Many well-known neurotransmitters including serotonin (5HT), dopamine (DA), histamine (HA), Gamma-Aminobutyric Acid (GABA) and acetylcholine (ACh) participate in volume transmission. Typically, the cell bodies are in one volume and the axons project to a distant volume in the brain releasing the neurotransmitter there. We introduce volume transmission and describe mathematically two natural homeostatic mechanisms. In some brain regions several neurotransmitters in the extracellular space affect each other's release. We investigate the dynamics created by this comodulation in two different cases: serotonin and histamine; and the comodulation of 4 neurotransmitters in the striatum and we compare to experimental data. This kind of comodulation poses new dynamical questions as well as the question of how these biochemical networks influence the electrophysiological networks in the brain. [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]

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]

The qualitative behavior of the differential equation x ′ = f (t , x) , where x is real and f is periodic in t, is determined by the attracting periodic solutions. This article gives an exposition of a method of Ilyashenko that uses complex analysis to determine an upper bound on the number of periodic solutions. The method is then applied to a well-known model of a single neuron or pool of neurons. [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]

The paper presents a conceptual mathematical model for Alzheimer's disease (AD). According to the so-called amyloid cascade hypothesis, we assume that the progression of AD is associated with the presence of soluble toxic oligomers of beta-amyloid. Monomers of this protein are produced normally throughout life, but a change in the metabolism may increase their total production and, through aggregation, ultimately results in a large quantity of highly toxic polymers. The evolution from monomeric amyloid produced by the neurons to senile plaques (long and insoluble polymeric amyloid chains) is modelled by a system of ordinary differential equations (ODEs), in the spirit of the Smoluchowski equation. The basic assumptions of the model are that, at the scale of suitably small representative elementary volumes (REVs) of the brain, the production of monomers depends on the average degradation of the neurons and in turn, at a much slower timescale, the degradation is caused by the number of toxic oligomers. To mimic prion-like diffusion of the disease in the brain, we introduce an interaction among adjacent REVs that can be assumed to be isotropic or to follow given preferential patterns. We display the results of numerical simulations which are obtained under some simplifying assumptions. For instance, the amyloid cascade is modelled by just three ordinary differential equations (ODEs), and the simulations refer to abstract 2D domains, simplifications which can be easily avoided at the price of some additional computational costs. Since the model is suitably flexible to incorporate other mechanisms and geometries, we believe that it can be generalised to describe more realistic situations. [ABSTRACT FROM AUTHOR]

By analyzing the Hodgkin-Huxley model, it can be estimated how neurons communicate with each other. It is the most widely used model that describes why the action potentials happen and how action potentials are generated and propagated in neurons. In a nerve cell, an action potential is evoked when the membrane potential attains its threshold (bifurcation point of neuron activity) membrane potential. The substantive changes associated with membrane voltage and conductance driving the action potential. Sometimes, nerve cells show abnormal activity for different causes. In this paper, we analyze the bifurcation of the Hodgkin-Huxley model to show the relationship between the abnormal activity of the nerve cells and the ionic conductance and external stimulation into the nerve cells. We also illustrated the possible region where the nerve cells show undesirable hyperactivity by calculating the Hopf bifurcation curve. [ABSTRACT FROM AUTHOR]

In this paper, a recurrent neural network with mixed delays which plays an important role is considered. We are concerned with the existence, uniqueness and global exponential stability of the doubly measure pseudo almost automorphic solutions. First, we establish results that are interesting on the functional space of such functions like composition theorem. Second, by employing the fixed-point theorem and some properties of the measure pseudo almost automorphic functions, some sufficient conditions for the existence, uniqueness and global exponential stability of the doubly measure pseudo almost automorphic solutions have been established. Our results obtained in this paper are new. Finally, two commonly used numerical examples are given to illustrate the effectiveness of the obtained results. [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]

Subthreshold oscillations in neurons are those oscillations which do not attain the critical value of the membrane's voltage needed for triggering an action potential (a spike). Their contribution to the forming of action potentials in neurons is a current field of research in biology. The present work approaches this subject using tools from mathematical modelling, more exactly, a neuronal non-smooth map-based model is proposed and studied. The behaviour of the model in a noisy medium is also studied. [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]

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 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]

Much work has been done to identify which binary codes can be represented by collections of open convex or closed convex sets. While not all binary codes can be realized by such sets, here we prove that every binary code can be realized by convex sets when there is no restriction on whether the sets are all open or closed. We achieve this by constructing a convex realization for an arbitrary code with k nonempty codewords in R k − 1 . This result justifies the usual restriction of the definition of convex neural codes to include only those that can be realized by receptive fields that are all either open convex or closed convex. We also show that the dimension of our construction cannot in general be lowered. [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 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]

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]

How does the brain encode spatial structure? One way is through hippocampal neurons called place cells, which become associated to convex regions of space known as their receptive fields: each place cell fires at a high rate precisely when the animal is in the receptive field. The firing patterns of multiple place cells form what is known as a convex neural code. How can we tell when a neural code is convex? To address this question, Giusti and Itskov identified a local obstruction, defined via the topology of a code's simplicial complex, and proved that convex neural codes have no local obstructions. Curto et al. proved the converse for all neural codes on at most four neurons. Via a counterexample on five neurons, we show that this converse is false in general. Additionally, we classify all codes on five neurons with no local obstructions. This classification is enabled by our enumeration of connected simplicial complexes on 5 vertices up to isomorphism. Finally, we examine how local obstructions are related to maximal codewords (maximal sets of neurons that co-fire). Curto et al. proved that a code has no local obstructions if and only if it contains certain “mandatory” intersections of maximal codewords. We give a new criterion for an intersection of maximal codewords to be non-mandatory, and prove that it classifies all such non-mandatory codewords for codes on up to five neurons. [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]

Exact, steady-state, single-front solutions are constructed for a spatially discrete bistable equation with a piecewise linear reaction term, known as a sawtooth nonlinearity. These solutions are obtained by solving second-order difference equations with variable coefficients, which are linear under certain assumptions on the expected solutions. An algorithmic procedure for constructing solutions in general, for both homogeneous and inhomogeneous diffusion, is obtained using a combination of Jacobi-Operator theory and the Sherman–Morrison formula. The existence of solutions for the difference equation, implies propagation failure of fronts for the corresponding differential-difference equation. The interval of propagation failure, which is the range of values of the detuning parameter that render stationary fronts, is studied in detail for the case of a single defect in the medium of propagation. Explicit formulae reveal precise relationships between parameter values that cause traveling fronts to fail to propagate when the interface reaches the inhomogeneities in the medium. These explicit formulae are also compared to numerical computations using the proposed algorithmic approach, which provides a check of its computational usefulness and illustrates its capabilities for problems with more complicated choices of parameter values. [ABSTRACT FROM PUBLISHER]

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]

We simulated the electric current distribution in the brain during transcranial direct current stimulation (tDCS) using an anatomically accurate human head model. We estimated an effect of common electrode montages on spatial distribution of the electric field during tDCS procedure and analyzed a sensitivity of the technique to variations of electrode size and orientation. We concluded that the used electrode montages are stable with respect to minor changes in electrode size and position, while an assumption of homogeneity and isotropy of the head model results in crucial changes of the electric field distribution. We determined the electrode montages suited to deliver strong effect on hippocampus and cerebellum. [ABSTRACT FROM AUTHOR]

We are concerned with the traveling wave solutions for the FitzHugh–Nagumo system on an infinite channel. Based on a variational formulation in which a non-local term depends on a parameter, the speed of a traveling wave can be selected out. Furthermore, to show the existence of a traveling wave solution with such a speed, we seek a minimizer subject to a constraint. In the way of solving the variational problem, we apply a truncation technique to the nonlocal term to obtain a minimizer located in a bounded invariant region. [ABSTRACT FROM AUTHOR]

In this paper we consider a bistable reaction-diffusion equation in unbounded domains and we investigate geometric conditions under which propagation, possibly partial, takes place in some direction or, on the contrary, there is a blocking phenomenon. We start by proving the well-posedness of the problem. Then we prove that when the domain has a decreasing cross section with respect to the direction of propagation there is complete propagation. Further, we prove that the wave can be blocked as it comes through an abrupt opening. Finally we discuss various general geometrical properties that ensure either partial or complete invasion by 1. In particular, we show that in a domain that is 'star-shaped' with respect to an axis, there is complete invasion by 1. [ABSTRACT FROM AUTHOR]

In this article, we study canard solutions of the forced van der Pol equation in the relaxation limit for low-, intermediate-, and high-frequency periodic forcing. A central numerical observation made herein is that there are two branches of canards in parameter space which extend across all positive forcing frequencies. In the low-frequency forcing regime, we demonstrate the existence of primary maximal canards induced by folded saddle nodes of type I and establish explicit formulas for the parameter values at which the primary maximal canards and their folds exist. Then, we turn to the intermediate- and high-frequency forcing regimes and show that the forced van der Pol possesses torus canards instead. These torus canards consist of long segments near families of attracting and repelling limit cycles of the fast system, in alternation. We also derive explicit formulas for the parameter values at which the maximal torus canards and their folds exist. Primary maximal canards and maximal torus canards correspond geometrically to the situation in which the persistent manifolds near the family of attracting limit cycles coincide to all orders with the persistent manifolds that lie near the family of repelling limit cycles. The formulas derived for the folds of maximal canards in all three frequency regimes turn out to be representations of a single formula in the appropriate parameter regimes, and this unification confirms the central numerical observation that the folds of the maximal canards created in the low-frequency regime continue directly into the folds of the maximal torus canards that exist in the intermediate- and high-frequency regimes. In addition, we study the secondary canards induced by the folded singularities in the low-frequency regime and find that the fold curves of the secondary canards turn around in the intermediate-frequency regime, instead of continuing into the high-frequency regime. Also, we identify the mechanism responsible for this turning. Finally, we show that the forced van der Pol equation is a normal form-type equation for a class of single-frequency periodically driven slow/fast systems with two fast variables and one slow variable which possess a non-degenerate fold of limit cycles. The analytic techniques used herein rely on geometric desingularisation, invariant manifold theory, Melnikov theory, and normal form methods. The numerical methods used herein were developed in Desroches et al. (SIAM J Appl Dyn Syst 7:1131-1162, , Nonlinearity 23:739-765 ). [ABSTRACT FROM AUTHOR]

Sensory contribution to perception and action depends on both sensory receptors and the organization of pathways (or projections) reaching the central nervous system. Unlike the semicircular canals that are divided into three discrete sensitivity directions, the utricle has a relatively complicated anatomical structure, including sensitivity directions over essentially $$360^{\circ }$$ of a curved, two-dimensional disk. The utricle is not flat, and we do not assume it to be. Directional sensitivity of individual utricular afferents decreases in a cosine-like fashion from peak excitation for movement in one direction to a null or near null response for a movement in an orthogonal direction. Directional sensitivity varies slowly between neighboring cells except within the striolar region that separates the medial from the lateral zone, where the directional selectivity abruptly reverses along the reversal line. Utricular primary afferent pathways reach the vestibular nuclei and cerebellum and, in many cases, converge on target cells with semicircular canal primary afferents and afference from other sources. Mathematically, some canal pathways are known to be characterized by symmetry groups related to physical space. These groups structure rotational information and movement. They divide the target neural center into distinct populations according to the innervation patterns they receive. Like canal pathways, utricular pathways combine symmetries from the utricle with those from target neural centers. This study presents a generic set of transformations drawn from the known structure of the utricle and therefore likely to be found in utricular pathways, but not exhaustive of utricular pathway symmetries. This generic set of transformations forms a 32-element group that is a semi-direct product of two simple abelian groups. Subgroups of the group include order-four elements corresponding to discrete rotations. Evaluation of subgroups allows us to functionally identify the spatial implications of otolith and canal symmetries regarding action and perception. Our results are discussed in relation to observed utricular pathways, including those convergent with canal pathways. Oculomotor and other sensorimotor systems are organized according to canal planes. However, the utricle is evolutionarily prior to the canals and may provide a more fundamental spatial framework for canal pathways as well as for movement. The fullest purely otolithic pathway is likely that which reaches the lumbar spine via Deiters' cells in the lateral vestibular nucleus. It will be of great interest to see whether symmetries predicted from the utricle are identified within this pathway. [ABSTRACT FROM AUTHOR]

Restricted Gauss–Markov processes are used to construct inhomogeneous leaky integrate-and-fire stochastic models for single neurons activity in the presence of a lower reflecting boundary and periodic input signals. The first-passage time problem through a time-dependent threshold is explicitly developed; numerical, simulation and asymptotic results for firing densities are provided. [ABSTRACT FROM AUTHOR]

We here consider a stochastic version of the classical neural field equation that is currently actively studied in the mathematical neuroscience community. Our goal is to present a well-known rigorous probabilistic framework in which to study these equations in a way that is accessible to practitioners currently working in the area, and thus to bridge some of the cultural/scientific gaps between probability theory and mathematical biology. In this way, the paper is intended to act as a reference that collects together relevant rigorous results about notions of solutions and well-posedness, which although may be straightforward to experts from SPDEs, are largely unknown in the neuroscientific community, and difficult to find in a very large body of literature. Moreover, in the course of our study we provide some new specific conditions on the parameters appearing in the equation (in particular on the neural field kernel) that guarantee the existence of a solution. [ABSTRACT FROM AUTHOR]

We introduce an extension of the classical neural field equation where the dynamics of the synaptic kernel satisfies the standard Hebbian type of learning (synaptic plasticity). Here, a continuous network in which changes in the weight kernel occurs in a specified time window is considered. A novelty of this model is that it admits synaptic weight decrease as well as the usual weight increase resulting from correlated activity. The resulting equation leads to a delay-type rate model for which the existence and stability of solutions such as the rest state, bumps, and traveling fronts are investigated. Some relations between the length of the time window and the bump width is derived. In addition, the effect of the delay parameter on the stability of solutions is shown. Also numerical simulations for solutions and their stability are presented. [ABSTRACT FROM AUTHOR]

Mathematical model describing signal transduction between neurons has been presented using the system of nonlinear ordinary differential equations. The Cauchy problem for the given system has been solved numerically and system parameters were adjusted to match the results of experimental measurements of extracellular postsynaptic potentials in rat hippocampus slices. While fitting model to the experimental data some values of synaptic parameters have been determined. [ABSTRACT FROM PUBLISHER]