Sensitivity to the cutoff value in the quadratic adaptive integrate-and-fire model

The quadratic adaptive integrate-and-fire model (Izhikecih 2003, 2007) is recognized as very interesting for its computational efficiency and its ability to reproduce many behaviors observed in cortical neurons. For this reason it is currently widely used, in particular for large scale simulations of neural networks. This model emulates the dynamics of the membrane potential of a neuron together with an adaptation variable. The subthreshold dynamics is governed by a two-parameter differential equation, and a spike is emitted when the membrane potential variable reaches a given cutoff value. Subsequently the membrane potential is reset, and the adaptation variable is added a fixed value called the spike-triggered adaptation parameter. We show in this note that when the system does not converge to an equilibrium point, both variables of the subthreshold dynamical system blow up in finite time whatever the parameters of the dynamics. The cutoff is therefore essential for the model to be well defined and simulated. The divergence of the adaptation variable makes the system very sensitive to the cutoff: changing this parameter dramatically changes the spike patterns produced. Furthermore from a computational viewpoint, the fact that the adaptation variable blows up and the very sharp slope it has when the spike is emitted implies that the time step of the numerical simulation needs to be very small (or adaptive) in order to catch an accurate value of the adaptation at the time of the spike. It is not the case for the similar quartic (Touboul 2008) and exponential (Brette and Gerstner 2005) models whose adaptation variable does not blow up in finite time, and which are therefore very robust to changes in the cutoff value

Propagation of chaos in neural fields

We consider the problem of the limit of bio-inspired spatially extended neuronal networks including an infinite number of neuronal types (space locations), with space-dependent propagation delays modeling neural fields. The propagation of chaos property is proved in this setting under mild assumptions on the neuronal dynamics, valid for most models used in neuroscience, in a mesoscopic limit, the neural-field limit, in which we can resolve the quite fine structure of the neuron's activity in space and where averaging effects occur. The mean-field equations obtained are of a new type: they take the form of well-posed infinite-dimensional delayed integro-differential equations with a nonlocal mean-field term and a singular spatio-temporal Brownian motion. We also show how these intricate equations can be used in practice to uncover mathematically the precise mesoscopic dynamics of the neural field in a particular model where the mean-field equations exactly reduce to deterministic nonlinear delayed integro-differential equations. These results have several theoretical implications in neuroscience we review in the discussion.Comment: Updated to correct an erroneous suggestion of extension of the results in Appendix B, and to clarify some measurability questions in the proof of Theorem

Macroscopic equations governing noisy spiking neuronal populations

At functional scales, cortical behavior results from the complex interplay of a large number of excitable cells operating in noisy environments. Such systems resist to mathematical analysis, and computational neurosciences have largely relied on heuristic partial (and partially justified) macroscopic models, which successfully reproduced a number of relevant phenomena. The relationship between these macroscopic models and the spiking noisy dynamics of the underlying cells has since then been a great endeavor. Based on recent mean-field reductions for such spiking neurons, we present here {a principled reduction of large biologically plausible neuronal networks to firing-rate models, providing a rigorous} relationship between the macroscopic activity of populations of spiking neurons and popular macroscopic models, under a few assumptions (mainly linearity of the synapses). {The reduced model we derive consists of simple, low-dimensional ordinary differential equations with} parameters and {nonlinearities derived from} the underlying properties of the cells, and in particular the noise level. {These simple reduced models are shown to reproduce accurately the dynamics of large networks in numerical simulations}. Appropriate parameters and functions are made available {online} for different models of neurons: McKean, Fitzhugh-Nagumo and Hodgkin-Huxley models

Heterogeneous connections induce oscillations in large scale networks

Realistic large-scale networks display an heterogeneous distribution of connectivity weights, that might also randomly vary in time. We show that depending on the level of heterogeneity in the connectivity coefficients, different qualitative macroscopic and microscopic regimes emerge. We evidence in particular generic transitions from stationary to perfectly periodic phase-locked regimes as the disorder parameter is increased, both in a simple model treated analytically and in a biologically relevant network made of excitable cells

On an explicit representation of the solution of linear stochastic partial differential equations with delays

Based on the analysis of a certain class of linear operators on a Banach space, we provide a closed form expression for the solutions of certain linear partial differential equations with non-autonomous input, time delays and stochastic terms, which takes the form of an infinite series expansion

Topological and Dynamical Complexity of Random Neural Networks

Random neural networks are dynamical descriptions of randomly interconnected neural units. These show a phase transition to chaos as a disorder parameter is increased. The microscopic mechanisms underlying this phase transition are unknown, and similarly to spin-glasses, shall be fundamentally related to the behavior of the system. In this Letter we investigate the explosion of complexity arising near that phase transition. We show that the mean number of equilibria undergoes a sharp transition from one equilibrium to a very large number scaling exponentially with the dimension on the system. Near criticality, we compute the exponential rate of divergence, called topological complexity. Strikingly, we show that it behaves exactly as the maximal Lyapunov exponent, a classical measure of dynamical complexity. This relationship unravels a microscopic mechanism leading to chaos which we further demonstrate on a simpler class of disordered systems, suggesting a deep and underexplored link between topological and dynamical complexity