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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 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]

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]

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]

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 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]

Epilepsy is among the most common serious disabling disorders of the brain, and the global burden of epilepsy exerts a tremendous cost to society. Most people with epilepsy have acquired forms, and the development of antiepileptogenic interventions could potentially prevent or cure these epilepsies [3,13]. The discovery of potential antiepileptogenic treatments is currently a high research priority. Clinical validation would require a means to identify populations of patients at particular high risk for epilepsy after a potential epileptogenic insult to know when to treat and to document prevention or cure. We investigate the development of post-traumatic epilepsy (PTE) following traumatic brain injury (TBI), because this condition offers the best opportunity to know the time of onset of epileptogenesis in patients. Epileptogenesis is common after TBI, and because much is known about the physical history of PTE, it represents a near-ideal human model in which to study the process of developing seizures. Using scalp and depth EEG recordings for six patients, the goal of our analysis is to find a way to quantitatively detect features in the EEG that could potentially help predict seizure onset post trauma. Unsupervised Diffusion Component Analysis [5], a novel approach based on the diffusion mapping framework [4], reduces data dimensionality and provides pattern recognition that can be used to distinguish different states of the patient, such as seizures and non-seizure spikes in the EEG. This method is also adapted to the data to enable the extraction of the underlying brain activity. Previous work has shown that such techniques can be useful for seizure prediction [6]. Some new results that demonstrate how this algorithm is used to detect spikes in the EEG data as well as other changes over time are shown. This nonlinear and local network approach has been used to determine if the early occurrences of specific electrical features of epileptogenesis, such as interictal epileptiform activity and morphologic changes in spikes and seizures, during the initial week after TBI predicts the development of PTE. [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]

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]

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]

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]

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]

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]

Travelling wave solutions of integro-differential equations for modeling one-dimensional neuronal networks, are studied. Under moderate continuity assumptions, necessary and sufficient conditions for the existence and uniqueness of monotone increasing (decreasing) Travelling wave solutions are established. Some faults in previous studies are corrected. [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]