Mathematical Neuroscience

Understanding the neural field activity for realistic living systems is a challenging task in contemporary neuroscience. Neural fields have been studied and developed theoretically and numerically with considerable success over the past four decades. However, to make effective use of such models, we need to identify their constituents in practical systems. This includes the determination of model parameters and in particular the reconstruction of the underlying effective connectivity in biological tissues.In this work, we provide an integral equation approach to the reconstruction of the neural connectivity in the case where the neural activity is governed by a delay neural field equation. As preparation, we study the solution of the direct problem based on the Banach fixed-point theorem. Then we reformulate the inverse problem into a family of integral equations of the first kind. This equation will be vector valued when several neural activity trajectories are taken as input for t...

The oculomotor delayed-response (ODR) task is a common experimental paradigm of working memory (WM) study, in which a monkey must fixate its gaze on the center of a screen and, following a brief cue that flashes on the screen, keep fixating for several more seconds before shifting its gaze to the location where the cue flashed. Consequently, in the delay period between the cue and the response the monkey must maintain a memory of cue location. Electrophysiological recordings from the prefrontal area of the cortex (PFC) revealed neurons that display selective persistent activity: their firing rate change induced by the cue persists through delay period, but only in response to a confined range of cue locations. This suggests that the representation of the cue is maintained in the network by a change in network activity profile. In this work, we study a network of rate-model neurons that is capable of preserving information about a past input, owing to structured connectivity and nonlinearities in the neuronal transfer function (TFs). Particularly, we focus on the acceleration of the TF close to firing threshold and the concavity around TF saturation. Any memory mechanism which exploits TF saturation means that some neurons must fire close to their saturation rates; with our model, however, we show that a certain relation between the excitatory and inhibitory neurons' TFs can cause an effective saturation in the network without forcing the neurons into the saturating parts of their TFs. In addition, this mechanism enables the erasure of memory at the end of the delay by a global excitatory signal. Finally, we demonstrate the mechanism in a model network of spiking neurons which describes with more detail the oscillatory dynamics in the state transition due to the interaction of membrane and synaptic time constants which is neglected in the rate model.

In a world where artificial intelligence is evolving rapidly, traditional education is in a crisis. The current system, by focusing on rote memorisation and the application of pre-defined skills, risks preparing learners for the world of yesterday, teaching them to be efficient cogs in a machine that is rapidly becoming automated. This paper introduces a novel theoretical framework based on the Unified Dynamic Model of Mind (UDMM) to address this existential challenge. Here, learning is understood as a continuous movement around a virtual attractor (W_{\text{ideal}}), which represents a comprehensive simulation of reality and embodied satisfaction. We propose that learning environments can either guide the internal model towards creative curiosity and cognitive independence, or deviate it into compulsive orbits driven by an external, coerced attractor (W_{\text{coerced}}). We demonstrate how Kullback-Leibler (KL) divergence and orbital dynamics can design supportive environments. Emotional aspects function as an error correction mechanism, while social aspects act as predictive scaffolds for coordination. This model offers a deeper understanding of learning differences, redefines success and motivation in the age of AI, and opens avenues for interventions targeting the cognitive attractor rather than overt behaviour. A supplementary appendix illustrates deviation mechanisms.

We introduce a test for robustness of heteroclinic cycles that appear in neural microcircuits modeled as coupled dynamical cells. Robust heteroclinic cycles (RHCs) can appear as robust attractors in Lotka-Volterra-type winnerless competition (WLC) models as well as in more general coupled and/or symmetric systems. It has been previously suggested that RHCs may be relevant to a range of neural activities, from encoding and binding to spatio-temporal sequence generation.The robustness or otherwise of such cycles depends both on the coupling structure and the internal structure of the neurons. We verify that robust heteroclinic cycles can appear in systems of three identical cells, but only if we require perturbations to preserve some invariant subspaces for the individual cells. On the other hand, heteroclinic attractors can appear robustly in systems of four or more identical cells for some symmetric coupling patterns, without restriction on the internal dynamics of the cells.

Contemporary models of addiction often struggle to explain compulsive and persistent behaviors that transcend the mere pursuit of reward, leading to a radical reshaping of an individual's motivations and values. In this paper, we introduce a novel computational framework based on the Unified Dynamic Model of the Mind (UDMM), which posits that the mind operates as a proactive inference system continuously aligning itself with a virtual internal compass, represented by an ideal probabilistic attractor (W_{ideal}). We conceptualize addiction as a "hijacking" of this attractor: the addictive substance or behavior, through hierarchical Bayesian learning and stochastic differential equations, updates the internal attractor. This leads to a migration of the attractor from a healthy, balanced center to a compulsive one that dominates behavioral dynamics. This transition explains phenomena such as craving and relapse as necessary consequences of a dynamical bifurcation in the state space. The discussion explores the potential for therapeutic interventions aimed at re-learning the healthy attractor and reshaping the internal landscape of values.

Current computational neuroscience models often struggle to account for the continuous, value-oriented human drive that is not satisfied by achieving discrete goals. This paper introduces a new conceptual framework, the Unified Dynamic Model of the Mind (UDMM), which posits the mind as an inferential system proactively shaping its environment. Central to UDMM is the concept of a "saturated world," reconceptualized here not as an achievable goal-state but as a virtual attractor-a hypothetical probabilistic distribution that cannot be fully attained, yet serves as a consistent compass guiding system behavior. We propose an initial mathematical formulation grounded in the Free Energy Principle and Active Inference (Friston, 2010), where the agent does not simply minimize prediction error to reach a stable state. Instead, it continuously minimizes the Kullback-Leibler (KL) Divergence between its expected future state distribution and the ideal distribution representing the saturated world. This approach produces behavior resembling a stable orbit around higher values. The paper outlines a roadmap for developing computational models based on this principle and opens new avenues for modeling meaning, purpose, and mental disorders as dysfunctions in the dynamic relationship with this virtual attractor.

The Hopfield recurrent neural network is a classical auto-associative model of memory, in which collections of symmetrically coupled McCulloch-Pitts binary neurons interact to perform emergent computation. Although previous researchers have explored the potential of this network to solve combinatorial optimization problems or store reoccurring activity patterns as attractors of its deterministic dynamics, a basic open problem is to design a family of Hopfield networks with a number of noise-tolerant memories that grows exponentially with neural population size. Here, we discover such networks by minimizing probability flow, a recently proposed objective for estimating parameters in discrete maximum entropy models. By descending the gradient of the convex probability flow, our networks adapt synaptic weights to achieve robust exponential storage, even when presented with vanishingly small numbers of training patterns. In addition to providing a new set of low-density error-correcting...

Neuronal growth cones navigate over long distances along specific pathways to find their correct targets. The mechanisms and molecules that direct this pathfinding are the topics of this review. Growth cones appear to be guided by at least four different mechanisms: contact attraction, chemoattraction, contact repulsion, and chemorepulsion. Evidence is accumulating that these mechanisms act simultaneously and in a coordinated manner to direct pathfinding and that they are mediated by mechanistically and evolutionarily conserved ligand-receptor systems.

In 1982 Teuvo Kohonen proposed an algorithm called Self Organizing Map (SOM) that is basically a stochastic model of the establishment of topology preserving connections between the retina and the cortex in our brain. That model has proved an efficient classification tool, a source of inspiration for other biological models and also for the understanding of the principles of self-organization. In this paper, we present a modification of the initial SOM to address an important problem in semantic cognition: the representation of human concepts in the brain. Much of the knowledge acquired in the last decades in this field comes from the study of patients with acquired difficulties in language. The modification presented here is inspired in semantic dementia, a devastating pathology that introduces the deterioration of semantic knowledge into the brain. The paper discusses the model and describes some preliminary results that show its interest as the basis of possible new tools to help to understand the considered type of diseases.

Understanding the neural field activity for realistic living systems is a challenging task in contemporary neuroscience. Neural fields have been studied and developed theoretically and numerically with considerable success over the past four decades. However, to make effective use of such models, we need to identify their constituents in practical systems. This includes the determination of model parameters and in particular the reconstruction of the underlying effective connectivity in biological tissues.In this work, we provide an integral equation approach to the reconstruction of the neural connectivity in the case where the neural activity is governed by a delay neural field equation. As preparation, we study the solution of the direct problem based on the Banach fixed-point theorem. Then we reformulate the inverse problem into a family of integral equations of the first kind. This equation will be vector valued when several neural activity trajectories are taken as input for t...

Understanding the neural field activity for realistic living systems is a challenging task in contemporary neuroscience. Neural fields have been studied and developed theoretically and numerically with considerable success over the past four decades. However, to make effective use of such models, we need to identify their constituents in practical systems. This includes the determination of model parameters and in particular the reconstruction of the underlying effective connectivity in biological tissues.In this work, we provide an integral equation approach to the reconstruction of the neural connectivity in the case where the neural activity is governed by a delay neural field equation. As preparation, we study the solution of the direct problem based on the Banach fixed-point theorem. Then we reformulate the inverse problem into a family of integral equations of the first kind. This equation will be vector valued when several neural activity trajectories are taken as input for t...

Nonlinear Noisy Leaky Integrate and Fire (NNLIF) models for neurons networks can be written as Fokker-Planck-Kolmogorov equations on the probability density of neurons, the main parameters in the model being the connectivity of the network and the noise. We analyse several aspects of the NNLIF model: the number of steady states, a priori estimates, blow-up issues and convergence toward equilibrium in the linear case. In particular, for excitatory networks, blow-up always occurs for initial data concentrated close to the firing potential. These results show how critical is the balance between noise and excitatory/inhibitory interactions to the connectivity parameter.AMS Subject Classification: 35K60, 82C31, 92B20.

We study a simplified model of the representation of colors in the primate primary cortical visual area V1. The model is described by an initial value problem related to a Hammerstein equation. The solutions to this problem represent the variation of the activity of populations of neurons in V1 as a function of space and color. The two space variables describe the spatial extent of the cortex while the two color variables describe the hue and the saturation represented at every location in the cortex. We prove the well-posedness of the initial value problem. We focus on its stationary, i.e. independent of time, and periodic in space solutions. We show that the model equation is equivariant with respect to the direct product G of the group of the Euclidean transformations of the planar lattice determined by the spatial periodicity and the group of color transformations, isomorphic to O(2), and study the equivariant bifurcations of its stationary solutions when some parameters in the model vary. Their variations may be caused by the consumption of drugs and the bifurcated solutions may represent visual hallucinations in space and color. Some of the bifurcated solutions can be determined by applying the Equivariant Branching Lemma (EBL) by determining the axial subgroups of G . These define bifurcated solutions which are invariant under the action of the corresponding axial subgroup. We compute analytically these solutions and illustrate them as color images. Using advanced methods of numerical bifurcation analysis we then explore the persistence and stability of these solutions when varying some parameters in the model. We conjecture that we can rely on the EBL to predict the existence of patterns that survive in large parameter domains but not to predict their stability. On our way we discover the existence of spatially localized stable patterns through the phenomenon of "snaking".

We introduce a test for robustness of heteroclinic cycles that appear in neural microcircuits modeled as coupled dynamical cells. Robust heteroclinic cycles (RHCs) can appear as robust attractors in Lotka-Volterra-type winnerless competition (WLC) models as well as in more general coupled and/or symmetric systems. It has been previously suggested that RHCs may be relevant to a range of neural activities, from encoding and binding to spatio-temporal sequence generation. The robustness or otherwise of such cycles depends both on the coupling structure and the internal structure of the neurons. We verify that robust heteroclinic cycles can appear in systems of three identical cells, but only if we require perturbations to preserve some invariant subspaces for the individual cells. On the other hand, heteroclinic attractors can appear robustly in systems of four or more identical cells for some symmetric coupling patterns, without restriction on the internal dynamics of the cells.

A major obstacle in the analysis of many physiological models is the issue of model simplification. Various methods have been used for simplifying such models, with one common technique being to eliminate certain 'fast' variables using a quasi-steady-state assumption. In this article, we show when such a physiological model reduction technique in a slow-fast system is mathematically justified. We provide counterexamples showing that this technique can give erroneous results near the onset of oscillatory behaviour which is, practically, the region of most importance in a model. In addition, we show that the singular limit of the first Lyapunov coefficient of a Hopf bifurcation in a slow-fast system is, in general, not equal to the first Lyapunov coefficient of the Hopf bifurcation in the corresponding layer problem, a seemingly counterintuitive result. Consequently, one cannot deduce, in general, the criticality of a Hopf bifurcation in a slow-fast system from the lower-dimensional layer problem.

We discuss the notion of excitability in 2D slow/fast neural models from a geometric singular perturbation theory point of view. We focus on the inherent singular nature of slow/fast neural models and define excitability via singular bifurcations. In particular, we show that type I excitability is associated with a novel singular Bogdanov-Takens/SNIC bifurcation while type II excitability is associated with a singular Andronov-Hopf bifurcation. In both cases, canards play an important role in the understanding of the unfolding of these singular bifurcation structures. We also explain the transition between the two excitability types and highlight all bifurcations involved, thus providing a complete analysis of excitability based on geometric singular perturbation theory.

A neural field models the large scale behaviour of large groups of neurons. We extend previous results for these models by including a diffusion term into the neural field, which models direct, electrical connections. We extend known and prove new sun-star calculus results for delay equations to be able to include diffusion and explicitly characterise the essential spectrum. For a certain class of connectivity functions in the neural field model, we are able to compute its spectral properties and the first Lyapunov coefficient of a Hopf bifurcation. By examining a numerical example, we find that the addition of diffusion suppresses non-synchronised steady-states while favouring synchronised oscillatory modes.

Period-doubling cascades for the quadratic family are familiar to every student of elementary dynamical systems. Cascades are also often observed in both numerical and scientific experiments. Yet in all but the simplest cases, very little is known about why they exist. When there is one cascade, there are almost always infinitely many. Further, there has been no systematic way of distinguishing the different types of cascades. In this paper, we explain why certain families of maps both in one dimension and higher dimensions have cascades and develop a method for counting them. The periods of the orbits in a cascade are {k, 2k, 4k, 8k. .. } for some k, and we refer to such a cascade as a period-k cascade. We refer to cascades which persist for all large parameters as unbounded cascades. We are able to enumerate the set of unbounded period-k cascades which occur for certain classes of families. The following example will give the flavor of our approach. Consider the map F (λ, x) = λ − x 2 + g(λ, x), where λ and x ∈ R, and g : R × R → R, and consider its (periodic) orbits as λ is varied. For every generic smooth C 1-bounded function g, the map F always has exactly the same number of period-k cascades as occur in the case g ≡ 0. That is, cascades are generically robust in the sense that for a residual set of g, there is no way to destroy them. We show that the results are not restricted to quadratic families, nor are they restricted to one dimension. We also give new topological results on the space of orbits in R × R N under the Hausdorff metric. We show that generically, after discarding "flip" orbits, each connected component is a one-manifold. Each unbounded component in this space of "nonflip" orbits must contain a cascade.

The elapsed time model has been widely studied in the context of mathematical neuroscience with many open questions left. The model consists of an age-structured equation that describes the dynamics of interacting neurons structured by the elapsed time since their last discharge. Our interest lies in highly connected networks leading to strong nonlinearities where perturbation methods do not apply. To deal with this problem, we choose a particular case which can be reduced to delay equations. We prove a general convergence result to a stationary state in the inhibitory and the weakly excitatory cases. Moreover, we prove the existence of particular periodic solutions with jump discontinuities in the strongly excitatory case. Finally, we present some numerical simulations which ilustrate various behaviors, which are consistent with the theoretical results.

Pituitary cells of the anterior pituitary gland secrete hormones in response to patterns of electrical activity. Several types of pituitary cells produce short bursts of electrical activity which are more effective than single spikes in evoking hormone release. These bursts, called pseudo-plateau bursts, are unlike bursts studied mathematically in neurons (plateau bursting) and the standard fast-slow analysis used for plateau bursting is of limited use. Using an alternative fast-slow analysis, with one fast and two slow variables, we show that pseudo-plateau bursting is a canard-induced mixed mode oscillation. Using this technique, it is possible to determine the region of parameter space where bursting occurs as well as salient properties of the burst such as the number of spikes in the burst. The information gained from this one-fast/twoslow decomposition complements the information obtained from a two-fast/one-slow decomposition.

Pituitary cells of the anterior pituitary gland secrete hormones in response to patterns of electrical activity. Several types of pituitary cells produce short bursts of electrical activity which are more effective than single spikes in evoking hormone release. These bursts, called pseudo-plateau bursts, are unlike bursts studied mathematically in neurons (plateau bursting) and the standard fast-slow analysis used for plateau bursting is of limited use. Using an alternative fast-slow analysis, with one fast and two slow variables, we show that pseudo-plateau bursting is a canard-induced mixed mode oscillation. Using this technique, it is possible to determine the region of parameter space where bursting occurs as well as salient properties of the burst such as the number of spikes in the burst. The information gained from this one-fast/two-slow decomposition complements the information obtained from a two-fast/one-slow decomposition.

The elapsed time model has been widely studied in the context of mathematical neuroscience with many open questions left. The model consists of an age-structured equation that describes the dynamics of interacting neurons structured by the elapsed time since their last discharge. Our interest lies in highly connected networks leading to strong nonlinearities where perturbation methods do not apply. To deal with this problem, we choose a particular case which can be reduced to delay equations. We prove a general convergence result to a stationary state in the inhibitory and the weakly excitatory cases. Moreover, we prove the existence of particular periodic solutions with jump discontinuities in the strongly excitatory case. Finally, we present some numerical simulations which ilustrate various behaviors, which are consistent with the theoretical results.

The so called Jacobian problem [1] or Jacobian conjecture [2], [3] demands the existence of inverse functions of polynomial nature when the Jacobian is a nonzero constant (=1). Bass, Connell, Wright in there paper [3] have shown that it is enough to construct (or show the existence of) such inverse polynomials for special type of cubic polynomials whose Jacobian is a nonzero constant (=1).To settle Jacobian conjecture one needs to show the existence of inverse polynomials for special type of cubic polynomials whose Jacobian is a nonzero constant (=1) for all dimensions [3]. In this paper we explicitly give these inverse polynomials for two and three dimensions, i.e. for two and three variables cases. Any higher dimensional cases are no different than these special cases and it is possible to obtain such inverse polynomials in any higher dimensional cases also.

A method is presented for the reduction of morphologically detailed microcircuit models to a point-neuron representation without human intervention. The simplification occurs in a modular workflow, in the neighborhood of a user specified network activity state for the reference model, the "operating point". First, synapses are moved to the soma, correcting for dendritic filtering by low-pass filtering the delivered synaptic current. Filter parameters are computed numerically and independently for inhibitory and excitatory input using a Green's function approach. Next, point-neuron models for each neuron in the microcircuit are fit to their respective morphologically detailed counterparts. Here, generalized integrate-and-fire point neuron models are used, leveraging a recently published fitting toolbox. The fits are constrained by currents and voltages computed in the morphologically detailed partner neurons with soma corrected synapses at three depolarizations about th...

The paper establishes an easily testable analytical criterion which is necessary and sufficient for the existence of supercritical Hopf bifurcations arbitrarily close to nominal neural networks with a symmetric neuron interconnection matrix. The importance of the result is discussed in relation with the fundamental issue of robustness of complete stability of symmetric neural networks with respect to errors in the electronic implementation.

Empirical studies have demonstrated that electrical activity of the neuron can directly affect neurite outgrowth. In this paper, we study the possible implications of activity-dependent neurite outgrowth for neuronal morphology and network development, using a model in which initially disconnected cells organize themselves into a network under the influence of their intrinsic activity. A neuron is modelled as a neuritic field, the growth of which depends on its own level of activity, and neurons become connected when their fields overlap. In a purely excitatory network, we have previously demonstrated that activity-dependent outgrowth in combination with a neuronal response function with some form of firing threshold is sufficient to cause a transient overproduction (overshoot) in the number of connections or synapses. Here we show that overshoot still takes place in a network of excitatory and inhibitory cells, and can even be enhanced. With delayed development of inhibition the growth curve of the number of inhibitory connections no longer exhibits overshoot. An interesting emergent property of the model is that, solely as the result of simple outgrowth rules and cell interactions, the (dendritic) fields of the inhibitory cells tend to become smaller than those of the excitatory cells, even if both type of cells have the same outgrowth properties. Other consequences of the interactions among outgrowth, excitation and inhibition are that (i) the spatial distribution of inhibitory cells becomes important in determining the level of inhibition; (ii) pruning of connections can no longer take place if the network has grown without proper electrical activity for longer than a certain critical period; (iii) inhibitory cells, by inducing outgrowth, can help to connect different parts of a structure. Further, the model predicts that excitatory cell death will be accompanied by an increased neuritic field of surviving neurons (''compensatory sprouting''). The similarities of the model with findings in developing tissue cultures of dissociated cells are extensively discussed.

In this paper, we consider the Kirchhoff plate equation with delay terms on the boundary control are added (see system (1.1) below). we give some instability examples of system (1.1) for some choices of delays. Finally, we prove its well-posedness, strong stability without any geometric condition and exponential stability under a multiplier geometric control condition.

In this paper, we consider the Kirchhoff plate equation with delay terms on the boundary control are added (see system (1.1) below). we give some instability examples of system (1.1) for some choices of delays. Finally, we prove its well-posedness, strong stability without any geometric condition and exponential stability under a multiplier geometric control condition.

The processes that develop and maintain the intrinsic electrical properties of neurons are modeled by allowing the maximal strength of membrane conductances to be slowly varying functions of the intracellular calcium concentration. The resulting dynamic regulation of conductances allows model neurons to self-assemble their membrane currents and to react to and recover from external perturbations. This dramatically increases the stability of the model neuron and makes its intrinsic characteristics activity-dependent. For example, model neurons in a two-cell network spontaneously differentiate in response to each other's activity. In a spatially extended model neuron, the dynamic regulatory mechanism causes a non-uniform distribution of currents to develop in response to both the morphology and the activity of the neuron.

We study pathwise approximation of strong solutions of scalar stochastic differential equations (SDEs) at a single time in the presence of discontinuities of the drift coefficient. Recently, it has been shown by Müller-Gronbach and Yaroslavtseva (2022) that for all p ∈ [1, ∞) a transformed Milstein-type scheme reaches an L p-error rate of at least 3/4 when the drift coefficient is a piecewise Lipschitz-continuous function with a piecewise Lipschitz-continuous derivative and the diffusion coefficient is constant. It has been proven by Müller-Gronbach and Yaroslavtseva (2023) that this rate 3/4 is optimal if one additionally assumes that the drift coefficient is bounded, increasing and has a point of discontinuity. While boundedness and monotonicity of the drift coefficient are crucial for the proof of the matching lower bound from Müller-Gronbach and Yaroslavtseva (2023), we show that both conditions can be dropped. For the proof we apply a transformation technique which was so far only used to obtain upper bounds.

To model the regulation of gene expression in eukaryotes by transcriptional activators and repressors, we introduce delays in conjugation with the mass action law. Delays are associated with the time gap between the mRNA transcription in the nucleoplasm and the protein synthesis in the cytoplasm. After re-parameterisation of the m-repressilator model with the Hill cooperative parameter n, for n = 1, the m-repressilator is deductible from the mass action law and, in the limit n → ∞, it is a Boolean type model. With this embedding and with delays, if m is odd and n > 1, we show that there is always a choice of parameters for which the m-repressilator model has sustained oscillations (limit cycles), implying that the 1-repressilator is the simplest genetic mechanism leading to sustained oscillations in eukaryotes. If m is even and n > 1, there is always a choice of parameters for which the m-repressilator model has bistability.

Understanding the dynamics of coupled neurons is one of the fundamental problems in the analysis of neuronal model dynamics. The transfer entropy (TE) method is one of the primary analyses to explore the information flow between the neuronal populations. We perform the TE analysis on the two-neuron conductance-based Hodgkin-Huxley (HH) neuronal network to analyze how their connectivity changes due to conductances. We find that the information flow due to underlying synaptic connectivity changes direction by changing conductances individually and/or simultaneously as a result of TE analysis through numerical simulations.

This paper investigates the asymptotical behavior of the equilibrium of linear classical duopolies by reconsidering the two-delay model with two different positive delays. In a two-dimensional analysis, the stability switching curves were first analytically determined. Numerical studies verified and illustrated the theoretical results. In the sensitivity analysis it was demonstrated that the inertia coefficient has a twofold effect: enlarges the stability region as well as simplifies the complicated dynamics with period-halving cascade. In contrary, the adjustment speed contracts the stability region and complicates simple dynamics with period-doubling bifurcation. In addition, for various values of τ1 and τ2, a wide variety of dynamics appears ranging from simple cycle via a Hopf bifurcation to chaotic oscillations.

Using a perturbative expansion for weak synaptic weights and weak sources of randomness, we calculate the correlation structure of neural networks with generic connectivity matrices. In detail, the perturbative parameters are the mean and the standard deviation of the synaptic weights, together with the standard deviations of the background noise of the membrane potentials and of their initial conditions. We also show how to determine the correlation structure of the system when the synaptic connections have a random topology. This analysis is performed on rate neurons described by Wilson and Cowan equations, since this allows us to find analytic results. Moreover, the perturbative expansion can be developed at any order and for a generic connectivity matrix. We finally show an example of application of this technique for a particular case of biologically relevant topology of the synaptic connections.

Perception seems so simple. I look out of the window to see houses, trees, people walking past, the sky above, the grass below. I hear birds in the trees, cars going past, the distant sound of an alarm. The world is full of objects that make their presence known to me through my senses-what could be more simple? Yet the efficacy of perceptual experience hides a host of questions for which we do not yet have the answers. Information reaching our senses is generally incomplete, ambiguous, distributed in space and time and not neatly sorted according to its source, so a key function of our perceptual systems is to discover the likely causes of our sensations. Perception as inference or hypothesis testing, formalised in the predictive coding theory, offers an attractive framework for exploring these issues. From this perspective, regularities or patterns provide perceptual systems with some traction, allowing the formation of expectations and a basis for decomposing the world into discrete objects. But in the dynamic world which we inhabit, object representations must be similarly dynamic, and need to form and dissolve, dominate and yield, in a way that facilitates veridical perception. In this talk I will discuss auditory scene analysis in the context of predictive coding using experimental data, exemplar models, and the phenomenon of perceptual multistability.

The dynamic behavior of n-firm oligopolies is examined without product differentiation and with linear price and cost functions. Continuous time scales are assumed with best response dynamics, in which case the equilibrium is asymptotically stable without delays. The firms are assumed to face both implementation and information delays. If the delays are equal, then the model is a single delay case, and the equilibrium is oscillatory stable if the delay is small, at the threshold stability is lost by Hopf bifurcation with cyclic behavior, and for larger delays, the trajectories show expanding cycles. In the case of the non-equal delays, the stability switching curves are constructed and the directions of stability switches are determined. In the case of growth rate dynamics, the local behavior of the trajectories is similar to that of the best response dynamics. Simulation studies verify and illustrate the theoretical findings.

Neuronal persistent activity has been primarily assessed in terms of electrical mechanisms, without attention to the complex array of molecular events that also control cell excitability. We developed a multiscale neocortical model proceeding from the molecular to the network level to assess the contributions of calcium (Ca 2+) regulation of hyperpolarization-activated cyclic nucleotide-gated (HCN) channels in providing additional and complementary support of continuing activation in the network. The network contained 776 compartmental neurons arranged in the cortical layers, connected using synapses containing AMPA/ NMDA/GABA A /GABA B receptors. Metabotropic glutamate receptors (mGluR) produced inositol triphosphate (IP 3) which caused the release of Ca 2+ from endoplasmic reticulum (ER) stores, with reuptake by sarco/ER Ca 2+-ATP-ase pumps (SERCA), and influence on HCN channels. Stimulus-induced depolarization led to Ca 2+ influx via NMDA and voltage-gated Ca 2+ channels (VGCCs). After a delay, mGluR activation led to ER Ca 2+ release via IP 3 receptors. These factors increased HCN channel conductance and produced firing lasting for $1 min. The model displayed inter-scale synergies among synaptic weights, excitation/ inhibition balance, firing rates, membrane depolarization, Ca 2+ levels, regulation of HCN channels, and induction of persistent activity. The interaction between inhibition and Ca 2+ at the HCN channel nexus determined a limited range of inhibition strengths for which intracellular Ca 2+ could prepare population-specific persistent activity. Interactions between metabotropic and ionotropic inputs to the neuron demonstrated how multiple pathways could contribute in a complementary manner to persistent activity. Such redundancy and complementarity via multiple pathways is a critical feature of biological systems. Mediation of activation at different time scales, and through different pathways, would be expected to protect against disruption, in this case providing stability for persistent activity.

Heteroclinic dynamics is a suitable framework to describe transient dynamics that is characteristic for ecological as well as neural systems, in particular for cognitive processes. We consider different heteroclinic networks and zoom into the dynamics that emerges right at different bifurcation points. We identify features of criticality such as a proliferation of the dynamical repertoire and slowing down of the dynamics at the very bifurcations and in their immediate vicinity. It qualifies these bifurcation points as candidates for working points in systems which store and transfer information. We observe the emergence of a heteroclinic cycle between three manifolds that are densely covered by periodic orbits. Which periodic orbits are selected sensitively depends on the initial conditions due to hidden conservation laws. Our results add to the general observation that emergent features may be expected at borderlines, here between different dynamical regimes in heteroclinic dynamics.

Acetylcholine (ACh) is secreted from cholinergic neurons in the basal forebrain to regions throughout the cerebral cortex, including the primary visual cortex (V1), and influences neuronal activities across all six layers via a form of diffuse extrasynaptic modulation termed volume transmission. To understand this effect in V1, we performed extracellular multipoint recordings of neuronal responses to drifting sinusoidal grating stimuli from the cortical layers of V1 in anesthetized rats and examined the modulatory effects of topically administered ACh. ACh facilitated or suppressed the visual responses of individual cells with a laminar bias: response suppression prevailed in layers 2/3, whereas response facilitation prevailed in layer 5. ACh effects on the stimulus contrast-response function showed that ACh changes the response gain upward or downward in facilitated or suppressed cells, respectively. Next, ACh effects on the signal-to-noise (S/N) ratio and the gratingphase information were tested. The grating-phase information was calculated as the F1/F0 ratio, which represents the amount of temporal response modulation at the fundamental frequency (F1) of a drifting grating relative to the mean evoked response (F0). In facilitated cells, ACh improved the S/N ratio, while in suppressed cells it enhanced the F1/F0 ratio without any concurrent reduction in the S/N ratio. These effects were predominantly observed in regular-spiking cells, but not in fast-spiking cells. Electrophysiological and histological findings suggest that ACh promotes the signaling of gratingphase information to higher-order areas by a suppressive effect on supragranular layers and enhances feedback signals with a high S/N ratio to subcortical areas by a facilitatory effect on infragranular layers. Thus, ACh distinctly and finely controls visual information processing in a manner that is specific for the modulation and cell type and is also laminar dependent.