Spatially structured oscillations in a two-dimensional excitatory neuronal network with synaptic depression

Abstract

We study the spatiotemporal dynamics of a two-dimensional excitatory neuronal network with synaptic depression. Coupling between populations of neurons is taken to be nonlocal, while depression is taken to be local and presynaptic. We show that the network supports a wide range of spatially structured oscillations, which are suggestive of phenomena seen in cortical slice experiments and in vivo. The particular form of the oscillations depends on initial conditions and the level of background noise. Given an initial, spatially localized stimulus, activity evolves to a spatially localized oscillating core that periodically emits target waves. Low levels of noise can spontaneously generate several pockets of oscillatory activity that interact via their target patterns. Periodic activity in space can also organize into spiral waves, provided that there is some source of rotational symmetry breaking due to external stimuli or noise. In the high gain limit, no oscillatory behavior exists, but a transient stimulus can lead to a single, outward propagating target wave.

Two-Dimensional Bumps in Piecewise Smooth Neural Fields with Synaptic Depression

“…When shift perturbations destabilize a bump, the resulting dynamics evolves to a traveling pulse solution. As we showed in previous work, synaptic depression is a reliable mechanism for generating traveling pulses in excitatory neural fields [19,20]. As illustrated in Figure 4.1, following a perturbation by a leftward shift, the bump initially expands and then starts to propagate.…”

“…However, there are well-known scenarios in neural field models with linear adaptation, where Hopf bifurcations can occur leading to spatially structured oscillations such as breathers, target patterns, and spiral waves [14,15,31,33,35]. Recently we have shown how spatially structured oscillations can also occur in neural fields with synaptic depression, provided that the firing rate function has finite gain [19,20]. It would be interesting to explore scenarios where oscillations arise in neural fields with synaptic depression and Heaviside nonlinearities via some form of generalized Hopf bifurcation, along lines analogous to recent studies of nonsmooth dynamical systems [10,44].…”

“…Synaptic depression is the process by which presynaptic resources such as chemical neurotransmitters or synaptic vesicles are depleted [45]. It can be incorporated into the scalar neural field model (1.1) by introducing a dynamic prefactor q in the nonlocal term according to [19,20] τ ∂u(x, t) ∂t…”

“…Hence β ∼ 0.0001 − 0.1 (ms) −1 for the given range of values of α. In previous work we focused on the role of synaptic depression in generating spatially structured oscillations in 1D [19] and 2D [20] excitatory neural networks with continuous firing rate functions. In the high-gain limit, oscillatory solutions no longer exist but stationary and traveling wave solutions can be constructed.…”

“…Moreover, following previous studies of 2D neural field models [20,25,26,29,35], we can transform system (3.1) into a fourthorder PDE, which is computationally less expensive to evaluate. Using the fact that the corresponding Hankel transform of…”