Course 6 Propagation of pulses in cortical networks: The single-spike approximation

Abstract

Abstract We study the propagation of traveling solitary pulses in one-dimensional cortical networks with two types of one-dimensional architectures: networks of excitatory neurons, and networks composed of both excitatory and inhibitory neurons. Each neuron is represented by the integrate-and-fire model, and is allowed to fire only one spike. The velocity and stability of propagating, continuous pulses are calculated analytically. While studying excitatory-only networks, we focus on the effects of synaptic and axonal delays. Two continuous pulses with different velocities exist if the synaptic coupling is larger than a minimal value; the pulse with the lower velocity is always unstable. Above a certain critical value of the constant delay, continuous pulses lose stability via a Hopf bifurcation, and lurching pulses with spatiotemporal periodicity emerge. The parameter regime for which lurching occurs is strongly affected by the synaptic footprint (connectivity) shape. A bistable regime, in which both continuous and lurching pulses can propagate, may occur with square footprint shapes but not with exponential footprint shapes. A perturbation calculation is used in order to calculate the spatial lurching period and the velocity of lurching pulses at large delay values. For strong synaptic coupling, the velocities of both the continuous pulse and the lurching pulse are governed by the tail of the synaptic footprint shape. We find analytically how the axonal propagation velocity reduces the velocity of continuous pulses; it does not affect the critical delay. We then focus on the propagation of traveling solitary pulses in networks of excitatory and inhibitory neurons. Two types of stable propagating pulses are observed. During fast pulses, inhibitory neurons fire a short time before or after the excitatory neurons. During slow pulses, inhibitory cells fire well before neighboring excitatory cells, and potentials of excitatory cells become negative and then positive before they fire. There is a bistable parameter regime in which both fast and slow pulses can propagate. Fast pulses can propagate at low levels of inhibition, are affected by fast excitation but are almost unaffected by slow excitation, and are easily elicited by stimulating groups of neurons. In contrast, slow pulses can propagate at intermediate levels of inhibition, and are difficult to evoke. They can propagate without slow excitation, but slow excitation makes their propagation substantially more robust. Strong inhibitory-to-inhibitory conductance eliminates the slow pulses and converts the fast traveling pulses into irregular pulses, in which the inhibitory neurons segregate into two groups, which have different firing delays with respect to their neighboring excitatory cells. Lurching pulses may propagate if the excitatory-to-excitatory coupling is slow and very strong. In general, the velocity of the fast pulse increases with the axonal conductance velocity c , but there are cases in which it decreases with c . We suggest that the fast and slow pulses observed in our model correspond to the fast and slow propagating activity observed in experiments on neocortical slices.