Traveling waves and pulses in a one-dimensional network of excitable integrate-and-fire neurons

We study the existence and stability of traveling waves and pulses in a one-dimensional network of integrate-and-fire neurons with synaptic coupling. This provides a simple model of excitable neural tissue. We first derive a self-consistency condition for the existence of traveling waves, which generates a dispersion relation between velocity and wavelength. We use this to investigate how wave-propagation depends on various parameters that characterize neuronal interactions such as synaptic and axonal delays, and the passive membrane properties of dendritic cables. We also establish that excitable networks support the propagation of solitary pulses in the long-wavelength limit. We then derive a general condition for the (local) asymptotic stability of traveling waves in terms of the characteristic equation of the linearized firing time map, which takes the form of an integro-difference equation of infinite order. We use this to analyze the stability of solitary pulses in the long-wavelength limit. Solitary wave solutions are shown to come in pairs with the faster (slower) solution stable (unstable) in the case of zero axonal delays; for non-zero delays and fast synapses the stable wave can itself destabilize via a Hopf bifurcation

Mean-field theory of globally coupled integrate-and-fire neural oscillators with dynamic synapses

We analyze the effects of synaptic depression or facilitation on the existence and stability of the splay or asynchronous state in a population of all-to-all, pulse-coupled neural oscillators. We use mean-field techniques to derive conditions for the local stability of the splay state and determine how stability depends on the degree of synaptic depression or facilitation. We also consider the effects of noise. Extensions of the mean-field results to finite networks are developed in terms of the nonlinear firing time map

Synaptically generated wave propagation in excitable neural media

We study the propagation of solitary waves in a one-dimensional network of excitable integrate-and-fire neurons with axo-dendritic synaptic coupling. We show that for small axonal delays there exists a stable solitary wave, and derive a power scaling law for the velocity as a function of the coupling. In the case of large axonal delays and fast synapses we establish that the solitary wave can destabilize via a Hopf bifurcation in the firing times

Synchrony in an array of integrate-and-fire neurons with dendritic structure

A one-dimensional array of pulse-coupled integrate-and-fire neurons, each filtering input through an idealized passive dendritic cable, is used to model the nonlinear behavior induced by axodendritic interactions in neural populations. The relative firing phase of the neurons in the array is derived in the weak-coupling regime. It is shown that for long-range excitatory coupling the phases can undergo a bifurcation from a synchronous state to a state of traveling oscillatory waves. We establish the possible role of dendritic structure in the desynchronization of cortical oscillations

Desynchronization, mode locking and bursting in strongly coupled integrate-and-fire oscillators

We show how a synchronized pair of integrate-and-fire neural oscillators with noninstantaneous synaptic interactions can destabilize in the strong coupling regime resulting in non-phase-locked behavior. In the case of symmetric inhibitory coupling, desynchronization produces an inhomogeneous state in which one of the oscillators becomes inactive (oscillator death). On the other hand, for asymmetric excitatory/inhibitory coupling, mode locking can occur leading to periodic bursting patterns. The consequences for large globally coupled networks is discussed

Solitary waves in a model of dendritic cable with active spines

We consider a continuum model of dendritic spines with active membrane dynamics uniformly distributed along a passive dendritic cable. Byconsidering a systematic reduction of the Hodgkin-Huxleydy namics that is valid on all but very short time-scales we derive 2 dimensional and 1 dimensional systems for excitable tissue, both of which may be used to model the active processes in spine-heads. In the first case the coupling of the spine head dynamics to a passive dendritic cable via a direct electrical connection yields a model that may be regarded as a simplification of the Baer and Rinzel cable theory of excitable spinynerv e tissue [3]. This model is computationally simple with few free parameters. Importantly, as in the full model, numerical simulation illustrates the possibilityof a traveling wave. We present a systematic numerical investigation of the speed and stability of the wave as a function of physiologically important parameters. A further reduction of this model suggests that active spine-head dynamics mayb e modeled byan all or none type process which we take to be of the integrate-and-fire (IF) type. The model is analytically tractable allowing the explicit construction of the shape of traveling waves as well as the calculation of wave speed as a function of system parameters. In general a slow and fast wave are found to co-exist. The behavior of the fast wave is found to closely reproduce the behavior of the wave seen in simulations of the more detailed model. Importantly a linear stability theory is presented showing that it is the faster of the two solutions that is stable. Beyond a critical value the speed of the stable wave is found to decrease as a function of spine density. Moreover, the speed of this wave is found to decrease as a function of the strength of the electrical resistor coupling the spine-head and the cable, such that beyond some critical value there is propagation failure. Finally we discuss the importance of a model of passive electrical cable coupled to a system of integrate-and-fire units for physiological studies of branching dendritic tissue with active spines

Spontaneous oscillations in a nonlinear delayed-feedback shunting model of the pupil light reflex

We analyze spontaneous oscillations in a second-order delayed-feedback shunting model of the pupil light reflex. This model describes in a simple fashion the nonlinear effects of both the iris and retinal parts of the reflex pathway. In the case of smooth negative feedback, linear stability analysis is used to determine the conditions for a Hopf bifurcation in the pupil area as a function of various neurophysiological parameters of the system such as the time delay and the strength of neural connections. We also investigate oscillation onset in the case of piecewise negative feedback and obtain an analytical expression for the period of oscillations. Finally, complex periodic behavior is shown to arise in the presence of mixed feedback

Saltatory waves in the spike-diffuse-spike model of active dendritic spines

In this Letter we present the explicit construction of a saltatory traveling pulse of non-constant profile in an idealized model of dendritic tissue. Excitable dendritic spine clusters, modeled with integrate-and-fire (IF) units, are connected to a passive dendritic cable at a discrete set of points. The saltatory nature of the wave is directly attributed to the breaking of translation symmetry in the cable. The conditions for propagation failure are presented as a function of cluster separation and IF threshold

Spike train dynamics underlying pattern formation in integrate-and-fire oscillator networks

A dynamical mechanism underlying pattern formation in a spatially extended network of integrate-and-fire oscillators with synaptic interactions is identified. It is shown how in the strong coupling regime the network undergoes a discrete Turing-Hopf bifurcation of the firing times from a synchronous state to a state with periodic or quasiperiodic variations of the interspike intervals on closed orbits. The separation of these orbits in phase space results in a spatially periodic pattern of mean firing rate across the network that is modulated by deterministic fluctuations of the instantaneous firing rate

Traveling waves in a chain of pulse-coupled oscillators

We derive conditions for the existence of traveling wave solutions in a chain of pulse-coupled integrate-and-fire oscillators with nearest-neighbor interactions and distributed delays. A linear stability analysis of the traveling waves is carried out in terms of perturbations of the firing times of the oscillators. It is shown how traveling waves destabilize when the detuning between oscillators or the strength of the coupling becomes too large