The Effect of Inhibitory Feedback in a Small Neural Network

Abstract

Since Hodgkin and Huxley presented their first mathematical model that describes the dynamics of the action potential in neurons, much research about brain functioning has been being done by mathematical scientists. This thesis is concerned with the effect of inhibitory feedback on the input/output relations of neurons. We study a small neuronal network in which one excitatory neuron feeds back onto itself through an inhibitory interneuron. Simulation is used to determine the firing rates in response to steady input current applied to the synapse point of the excitatory neuron with different feedback strengths. The effect of feedback is to linearize at least part of the input/output relation (Mascagni, 1987). Results are compared to those of a reduced steady-state model in which the neurons are replaced by their open-loop input-output relationships and the closed-loop behavior is then predicted by solving an algebraic equation.

The model neurons in our simulations are spatially distributed. Each of them has one dendrite, a cell body, and an axon that makes a synapse onto the dendrite of the other neuron, at a location that is far enough away from the cell body to avoid the influence of spike generation on synaptic current. Synaptic dynamics are governed by the dynamics of presynaptic calcium current and postsynaptic conductance. The postsynaptic conductance dynamics are fast for the excitatory synapse and slow for the inhibitory synapse. The axonal conductances are those of a Traub neuron (Type I). Our simulation methodology is second-order accurate in space and time, even at junctions. We find that the firing rate as a function of input current converges to a linear relationship as the feedback strength increases, and that this result is well predicted by the reduced model.