Stefanos E. Folias

Breathing Pulses in an Excitatory Neural Network

In this paper we show how a local inhomogeneous input can stabilize a stationary-pulse solution in an excitatory neural network. A subsequent reduction of the input amplitude can then induce a Hopf instability of the stationary solution resulting in the formation of a breather. The breather can itself undergo a secondary instability leading to the periodic emission of traveling waves. In one dimension such waves consist of pairs of counterpropagating pulses, whereas in two dimensions the waves are circular target patterns.

Front bifurcations in an excitatory neural network

We show how a one-dimensional excitatory neural network can exhibit a symmetry breaking front bifurcation analogous to that found in reaction diffusion systems. This occurs in a homogeneous network when a stationary front undergoes a pitchfork bifurcation leading to bidirectional wave propagation. We analyze the dynamics in a neighborhood of the front bifurcation using perturbation methods, and we establish that a weak input inhomogeneity can induce a Hopf instability of the stationary front, leading to the formation of an oscillatory front or breather. We then carry out a stability analysis of stationary fronts in an exactly solvable model and use this to derive conditions for oscillatory fronts beyond the weak input regime. In particular, we show how wave propagation failure occurs in the presence of a large stationary input due to the pinning of a stationary front; a subsequent reductionin the strength of the input then generates a breather via a Hopf instability of the front. Finally, we derive condit...

Traveling pulses and wave propagation failure in inhomogeneous neural media

We use averaging and homogenization theory to study the propagation of traveling pulses in an inhomogeneous excitable neural network. The network is modeled in terms of a nonlocal integro- differential equation, in which the integral kernel represents the spatial distribution of synaptic weights. We show how a spatially periodic modulation of homogeneous synaptic connections leads to an effective reduction in the speed of a traveling pulse. In the case of large amplitude modulations, the traveling pulse represents the envelope of a multibump solution, in which individual bumps are nonpropagating and transient. The appearance (disappearance) of bumps at the leading (trailing) edge of the pulse generates the coherent propagation of the pulse. Wave propagation failure occurs when activity is insufficient to maintain bumps at the leading edge.

Stimulus-locked traveling waves and breathers in an excitatory neural network

We analyze the existence and stability of stimulus-locked traveling waves in a one-dimensional synaptically coupled excitatory neural network. The network is modeled in terms of a nonlocal integro-differential equation, in which the integral kernel represents the spatial distribution of synaptic weights, and the output firing rate of a neuron is taken to be a Heaviside function of activity. Given an inhomogeneous moving input of amplitude I0 and velocity v, we derive conditions for the existence of stimulus-locked waves by working in the moving frame of the input. We use this to construct existence tongues in (v,I0 )-parameter space whose tips at I0 = 0 correspond to the intrinsic waves of the homogeneous network. We then determine the linear stability of stimulus-locked waves within the tongues by constructing the associated Evans function and numerically calculating its zeros as a function of network parameters. We show that, as the input amplitude is reduced, a stimulus-locked wave within the tongue of...

Nonlinear Analysis of Breathing Pulses in a Synaptically Coupled Neural Network

We analyze the weakly nonlinear stability of a stationary pulse undergoing a Hopf bifurcation in a neural field model with an excitatory or Mexican hat synaptic weight function and Heaviside firing rate nonlinearity. The presence of a spatially localized input inhomogeneity

Spatiotemporal Pattern Formation in Neural Fields with Linear Adaptation

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Bifurcations of Stationary Solutions in an Interacting Pair of E-I Neural Fields ∗

precludes the 0 eigenvalue related to translation invariance of the pulse. Consequently, in the spectral analysis of the linearization about the stationary pulse

Traveling waves and breathers in an excitatory-inhibitory neural field

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Spatially Localized Synchronous Oscillations in Synaptically Coupled Neuronal Networks: Conductance-based Models and Discrete Maps

, there are two spatial modes, either of which can undergo a Hopf bifurcation in the Mexican hat network to produce a periodic orbit that either expands/contracts (breather) or moves side-to-side (slosher). We derive the normal form for each mode becoming critical in the Hopf bifurcation by (i) the method of amplitude equations and (ii) center manifold reduction, which are shown to agree. Importantly, the critical third order coefficient of the normal form is found to be in strong agreement with numerical simulations of the full model, particularly when...

Breathers in two-dimensional neural media.

We study spatiotemporal patterns of activity that emerge in neural fields in the presence of linear adaptation . Using an amplitude equation approach, we show that bifurcations from the homogeneous rest state can lead to a wide variety of stationary and propagating patterns on one- and two-dimensional periodic domains, particularly in the case of lateral-inhibitory synaptic weights. Other typical solutions are stationary and traveling localized activity bumps ; however, we observe exotic time-periodic localized patterns as well. Using linear stability analysis that perturbs about stationary and traveling bump solutions, we study conditions for the activity to lock to a stationary or traveling external input on both periodic and infinite one-dimensional spatial domains. Hopf and saddle-node bifurcations can signify the boundary beyond which stationary or traveling bumps fail to lock to external inputs. Just beyond a Hopf bifurcation point, activity bumps often begin to oscillate, becoming breather or slosher solutions.