On Local Bifurcations in Neural Field Models with Transmission Delays

Neural field models with transmission delay may be cast as abstract delay differential equations (DDE). The theory of dual semigroups (also called sun-star calculus) provides a natural framework for the analysis of a broad class of delay equations, among which DDE. In particular, it may be used advantageously for the investigation of stability and bifurcation of steady states. After introducing the neural field model in its basic functional analytic setting and discussing its spectral properties, we elaborate extensively an example and derive a characteristic equation. Under certain conditions the associated equilibrium may destabilise in a Hopf bifurcation. Furthermore, two Hopf curves may intersect in a double Hopf point in a two-dimensional parameter space. We provide general formulas for the corresponding critical normal form coefficients, evaluate these numerically and interpret the results

Lumping Izhikevich neurons

We present the construction of a planar vector field that yields the firing rate of a bursting Izhikevich neuron can be read out, while leaving the sub-threshold behaviour intact. This planar vector field is used to derive lumped formulations of two complex heterogeneous networks of bursting Izhikevich neurons. In both cases, the lumped model is compared with the spiking network. There is excellent agreement in terms of duration and number of action potentials within the bursts, but there is a slight mismatch of the burst frequency. The lumped model accurately accounts for both intrinsic bursting and post inhibitory rebound potentials in the neuron model, features which are absent in prevalent neural mass models

Comparing Epileptiform Behavior of Mesoscale Detailed Models and Population Models of Neocortex

Two models of the neocortex are developed to study normal and pathologic neuronal activity. One model contains a detailed description of a neocortical microcolumn represented by 656 neurons, including superficial and deep pyramidal cells, four types of inhibitory neurons, and realistic synaptic contacts. Simulations show that neurons of a given type exhibit similar, synchronized behavior in this detailed model. This observation is captured by a population model that describes the activity of large neuronal populations with two differential equations with two delays. Both models appear to have similar sensitivity to variations of total network excitation. Analysis of the population model reveals the presence of multistability, which was also observed in various simulations of the detailed model

Large-Scale Modeling of Epileptic Seizures: Scaling Properties of Two Parallel Neuronal Network Simulation Algorithms

Our limited understanding of the relationship between the behavior of individual neurons and large neuronal networks is an important limitation in current epilepsy research and may be one of the main causes of our inadequate ability to treat it. Addressing this problem directly via experiments is impossibly complex; thus, we have been developing and studying medium-large-scale simulations of detailed neuronal networks to guide us. Flexibility in the connection schemas and a complete description of the cortical tissue seem necessary for this purpose. In this paper we examine some of the basic issues encountered in these multiscale simulations. We have determined the detailed behavior of two such simulators on parallel computer systems. The observed memory and computation-time scaling behavior for a distributed memory implementation were very good over the range studied, both in terms of network sizes (2,000 to 400,000 neurons) and processor pool sizes (1 to 256 processors). Our simulations required between a few megabytes and about 150 gigabytes of RAM and lasted between a few minutes and about a week, well within the capability of most multinode clusters. Therefore, simulations of epileptic seizures on networks with millions of cells should be feasible on current supercomputers

Standing and travelling waves in a spherical brain model: the Nunez model revisited

International audienceThe Nunez model for the generation of electroencephalogram (EEG) signals is naturally described as a neural field model on a sphere with space-dependent delays. For simplicity, dynamical realisations of this model either as a damped wave equation or an integro-differential equation, have typically been studied in idealised one dimensional or planar settings. Here we revisit the original Nunez model to specifically address the role of spherical topology on spatio-temporal pattern generation. We do this using a mixture of Turing instability analysis, symmetric bifurcation theory, center manifold reduction and direct simulations with a bespoke numerical scheme. In particular we examine standing and travelling wave solutions using normal form computation of primary and secondary bifurcations from a steady state. Interestingly, we observe spatio-temporal patterns which have counterparts seen in the EEG patterns of both epileptic and schizophrenic brain conditions

Analysis of stability and bifurcations of fixed points and periodic solutions of a lumped model of neocortex with two delays

A lumped model of neural activity in neocortex is studied to identify regions of multi-stability of both steady states and periodic solutions. Presence of both steady states and periodic solutions is considered to correspond with epileptogenesis. The model, which consists of two delay differential equations with two fixed time lags is mainly studied for its dependency on varying connection strength between populations. Equilibria are identified, and using linear stability analysis, all transitions are determined under which both trivial and non-trivial fixed points lose stability. Periodic solutions arising at some of these bifurcations are numerically studied with a two-parameter bifurcation analysis