Conditions for wave trains in spiking neural networks

Abstract

Spatiotemporal patterns such as traveling waves are frequently observed in recordings of neural activity. The mechanisms underlying the generation of such patterns are largely unknown. Previous studies have investigated the existence and uniqueness of different types of waves or bumps of activity using neural-field models, phenomenological coarse-grained descriptions of neural-network dynamics. But it remains unclear how these insights can be transferred to more biologically realistic networks of spiking neurons, where individual neurons fire irregularly. Here, we employ mean-field theory to reduce a microscopic model of leaky integrate-and-fire (LIF) neurons with distance-dependent connectivity to an effective neural-field model. In contrast to existing phenomenological descriptions, the dynamics in this neural-field model depends on the mean and the variance in the synaptic input, both determining the amplitude and the temporal structure of the resulting effective coupling kernel. For the neural-field model we employ liner stability analysis to derive conditions for the existence of spatial and temporal oscillations and wave trains, that is, temporally and spatially periodic traveling waves. We first prove that wave trains cannot occur in a single homogeneous population of neurons, irrespective of the form of distance dependence of the connection probability. Compatible with the architecture of cortical neural networks, wave trains emerge in two-population networks of excitatory and inhibitory neurons as a combination of delay-induced temporal oscillations and spatial oscillations due to distance-dependent connectivity profiles. Finally, we demonstrate quantitative agreement between predictions of the analytically tractable neural-field model and numerical simulations of both networks of nonlinear rate-based units and networks of LIF neurons.

Boundary solution based on rescaling method: recoup the first and second-order statistics of neuron network dynamics

“…The topographic pattern of connection is interrupted in the network edges, changing the activity in the network boundary [14,15]. A classic solution adopted to this problem is the torus connection, which introduces undesired oscillations to the network [22,21,25].…”

Diversity of neuronal activity is provided by hybrid synapses

“…The combination of multiple traveling waves can form complex spatiotemporal patterns [4,47]. It has been shown, in spiking neural networks [46,[48][49][50][51]that transmission delays [41], the spatial reach of connections [7] and the strength of inhibition of the excitatory-inhibitory networks influence the emergence of spatiotemporal patterns, such as asynchronous and irregular activity, or propagating waves. Most previous modeling studies have shown that the formation of spatiotemporal patterns such as traveling waves in neural networks has mainly been investigated by means of phenomenological neural-field models.…”

Dynamics of neural fields with exponential temporal kernel

“…The objective of this work is to analytically and numerically study the static and dynamic bifurcations and spatio-temporal wave patterns generated by the classical neural field model with an exponential temporal kernel. This form of the temporal kernel is more general than the Green's function used in [16] and [20]. In [20] the temporal connectivity kernel is the product of an alpha function and the Heaviside function, which yields a function with the same properties as the Green's function, and thus yields the same characteristic polynomial as in [16].…”

“…This form of the temporal kernel is more general than the Green's function used in [16] and [20]. In [20] the temporal connectivity kernel is the product of an alpha function and the Heaviside function, which yields a function with the same properties as the Green's function, and thus yields the same characteristic polynomial as in [16]. We recall that the Green's function G(t, t ) is the solution to LG(t, t ) = δ(t−t ) satisfying the given boundary conditions, where L is a differential operator.…”

“…Ref. [20] is quite inspiring for reducing a biologically more realistic microscopic model of leaky integrate-and-fire neurons with distance-dependent connectivity to an effective neural field model. Because of the type of kernels used there, two different neuron populations, excitatory and inhibitory ones, are needed to induce dynamic bifurcations.…”