In this paper we deal with a model describing the evolution in time of the density of aneural population in a state space, where the state is given by Izhikevich’s two -dimensional single neuron model. The main goal is to mathematically describe theoccurrence of a significant phenomenon observed in neurons populations, thesynchronization. To this end, we are making the transition to phasedensity population, and use Malkin theorem to calculate the phase deviations of a weaklycoupled population model.

We analyze a stochastic neuronal network model which corresponds to an all-to-all networkof discretized integrate-and-fire neurons where the synapses are failure-prone. Thisnetwork exhibits different phases of behavior corresponding to synchrony and asynchrony,and we show that this is due to the limiting mean-field system possessing multipleattractors. We also show that this mean-field limit exhibits a first-order phasetransition as a function of the connection strength — as the synapses are made morereliable, there is a sudden onset of synchronous behavior. A detailed understanding of thedynamics involves both a characterization of the size of the giant component in a certainrandom graph process, and control of the pathwise dynamics of the system by obtainingexponential bounds for the probabilities of events far from the mean.

We study the coexistence of multiple periodic solutions for an analogue of theintegrate-and-fire neuron model of two-neuron recurrent inhibitory loops with delayedfeedback, which incorporates the firing process and absolute refractory period. Uponreceiving an excitatory signal from the excitatory neuron, the inhibitory neuron emits aspike with a pattern-related delay, in addition to the synaptic delay. We present atheoretical framework to view the inhibitory signal from the inhibitory neuron as aself-feedback of the excitatory neuron with this additional delay. Our analysis shows thatthe inhibitory feedbacks with firing and the absolute refractory period can generate fourbasic types of oscillations, and the complicated interaction among these basicoscillations leads to a large class of periodic patterns and the occurrence ofmultistability in the recurrent inhibitory loop. We also introduce the average time ofconvergence to a periodic pattern to determine which periodic patterns have the potentialto be used for neural information transmission and cognition processing in the nervoussystem.

The interplay between intrinsic and network dynamics has been the focus of manyinvestigations. Here we use a combination of theoretical and numerical approaches to studythe effects of delayed global feedback on the information transmission properties ofneural networks. Specifically, we compare networks of neurons that display intrinsicinterspike interval correlations (nonrenewal) to networks that do not (renewal). We findthat excitatory and inhibitory delays can tune information transmission by single neuronsbut not by the entire network. Most surprisingly, addition of a delay can change thedependence of the information on the coupling strength for renewal neurons and not fornonrenewal neurons. Our results show that intrinsic ISI correlations can have nontrivialinteractions with network-induced phenomena.

Computational models for human decision making are typically based on the properties of bistable dynamical systems where each attractor represents a different decision. A limitation of these models is that they do not readily account for the fragilities of human decision making, such as “choking under pressure”, indecisiveness and the role of past experiences on current decision making. Here we examine the dynamics of a model of two interacting neural populations with mutual time–delayed inhibition. When the input to each population is sufficiently high, there is bistability and the dynamics is determined by the relationship of the initial function to the separatrix (the stable manifold of a saddle point) that separates the basins of attraction of two co–existing attractors. The consequences for decision making include long periods of indecisiveness in which trajectories are confined in the neighborhood of the separatrix and wrong decision making, particularly when the effects of past history and irrelevant information (“noise”) are included. Since the effects of delay, past history and noise on bistable dynamical systems are generic, we anticipate that similar phenomena will arise in the setting of other physical, chemical and neural time–delayed systems which exhibit bistability.

We consider the problem of state and parameter estimation for a class of nonlinearoscillators defined as a system of coupled nonlinear ordinary differential equations.Observable variables are limited to a few components of state vector and an input signal.This class of systems describes a set of canonic models governing the dynamics of evokedpotential in neural membranes, including Hodgkin-Huxley, Hindmarsh-Rose, FitzHugh-Nagumo,and Morris-Lecar models. We consider the problem of state and parameter reconstruction forthese models within the classical framework of observer design. This framework offerscomputationally-efficient solutions to the problem of state and parameter reconstructionof a system of nonlinear differential equations, provided that these equations are in theso-called adaptive observer canonic form. We show that despite typical neural oscillatorsbeing locally observable they are not in the adaptive canonic observer form. Furthermore,we show that no parameter-independent diffeomorphism exists such that the originalequations of these models can be transformed into the adaptive canonic observer form. Wedemonstrate, however, that for the class of Hindmarsh-Rose and FitzHugh-Nagumo models,parameter-dependent coordinate transformations can be used to render these systems intothe adaptive observer canonical form. This allows reconstruction, at least partially andup to a (bi)linear transformation, of unknown state and parameter values with exponentialrate of convergence. In order to avoid the problem of only partial reconstruction and atthe same time to be able to deal with more general nonlinear models in which the unknownparameters enter the system nonlinearly, we present a new method for state and parameterreconstruction for these systems. The method combines advantages of standardLyapunov-based design with more flexible design and analysis techniques based on thenotions of positive invariance and small-gain theorems. We show that this flexibilityallows to overcome ill-conditioning and non-uniqueness issues arising in this problem.Effectiveness of our method is illustrated with simple numerical examples.

Recent technological advances including brain imaging (higher resolution in space andtime), miniaturization of integrated circuits (nanotechnologies), and acceleration ofcomputation speed (Moore’s Law), combined with interpenetration between neuroscience,mathematics, and physics have led to the development of more biologically plausiblecomputational models and novel therapeutic strategies. Today, mathematical models ofirreversible medical conditions such as Parkinson’s disease (PD) are developed andparameterised based on clinical data. How do these evolutions have a bearing on deep brainstimulation (DBS) of patients with PD? We review how the idea of DBS, a standardtherapeutic strategy used to attenuate neurological symptoms (motor, psychiatric), hasemerged from past experimental and clinical observations, and present how, over the lastdecade, computational models based on different approaches (phase oscillator models,spiking neuron network models, population-based models) have started to shed light ontoDBS mechanisms. Finally, we explore a new mathematical modelling approach based on neuralfield equations to optimize mechanisms of brain stimulation and achieve finer control oftargeted neuronal populations. We conclude that neuroscience and mathematics are crucialpartners in exploring brain stimulation and this partnership should also include otherdomains such as signal processing, control theory and ethics.