Abstract
We consider the constrained optimization of excitatory synaptic input patterns to maximize spike generation in leaky integrate-and-fire (LIF) and theta model neurons. In the case of discrete input kicks with a fixed total magnitude, optimal input timings and strengths are identified for each model using phase plane arguments. In both cases, optimal features relate to finding an input level at which the drop in input between successive spikes is minimized. A bounded minimizing level always exists in the theta model and may or may not exist in the LIF model, depending on parameter tuning. We also provide analytical formulas to estimate the number of spikes resulting from a given input train. In a second case of continuous inputs of fixed total magnitude, we analyze the tuning of an input shape parameter to maximize the number of spikes occurring in a fixed time interval. Results are obtained using numerical solution of a variational boundary value problem that we derive, as well as analysis, for the theta model and using a combination of simulation and analysis for the LIF model. In particular, consistent with the discrete case, the number of spikes in the theta model rises and then falls again as the input becomes more tightly peaked. Under a similar variation in the LIF case, we numerically show that the number of spikes increases monotonically up to some bound and we analytically constrain the times at which spikes can occur and estimate the bound on the number of spikes fired.
Highlights
A major component of theoretical neuroscience is the study of how various neuronal models respond to synaptic inputs
We find that the phase plane structures for the theta model resemble one of the possible cases for the leaky integrate-and-fire (LIF) model, with a corresponding similarity in optimal strategies
We have considered how certain constrained, positive inputs should be timed to yield maximal numbers of spikes in the LIF and theta models for neurons
Summary
A major component of theoretical neuroscience is the study of how various neuronal models respond to synaptic inputs. Equations of the form (18)-(21), each requiring calculation of only a small number of quantities, can be used on a case by case basis to estimate the numbers of spikes that will result from a given strategy and to compare strategies These formulas provide for an informed comparison between the two types of big kick strategies determined to be optimal for the two distinct cases of E + I − 2EI ≥ 0 and E + I − 2EI < 0, respectively, and the critical kicks strategy. That we have defined δ(g), we can give a more precise variation on the calculation of equation (9) made in the subsection on phase plane structures and basic strategies to show that truly optimal strategies (other than the critical kick strategy based on γc) provide kicks with v = I , so each strategy should include a time shift so that kicks are given when this condition is met, rather than with v = 0 or v = 1. It is not difficult to see from examination of the above spike counts and equation (10) that, with other parameters fixed, increases in β yield fewer spikes, as expected from the corresponding faster decay of g, while increases in E and I yield more spikes, as expected from the increased rate of change of v
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