Abstract
Objective. External control of spike times in single neurons can reveal important information about a neuron's sub-threshold dynamics that lead to spiking, and has the potential to improve brain–machine interfaces and neural prostheses. The goal of this paper is the design of optimal electrical stimulation of a neuron to achieve a target spike train under the physiological constraint to not damage tissue. Approach. We pose a stochastic optimal control problem to precisely specify the spike times in a leaky integrate-and-fire (LIF) model of a neuron with noise assumed to be of intrinsic or synaptic origin. In particular, we allow for the noise to be of arbitrary intensity. The optimal control problem is solved using dynamic programming when the controller has access to the voltage (closed-loop control), and using a maximum principle for the transition density when the controller only has access to the spike times (open-loop control). Main results. We have developed a stochastic optimal control algorithm to obtain precise spike times. It is applicable in both the supra-threshold and sub-threshold regimes, under open-loop and closed-loop conditions and with an arbitrary noise intensity; the accuracy of control degrades with increasing intensity of the noise. Simulations show that our algorithms produce the desired results for the LIF model, but also for the case where the neuron dynamics are given by more complex models than the LIF model. This is illustrated explicitly using the Morris–Lecar spiking neuron model, for which an LIF approximation is first obtained from a spike sequence using a previously published method. We further show that a related control strategy based on the assumption that there is no noise performs poorly in comparison to our noise-based strategies. The algorithms are numerically intensive and may require efficiency refinements to achieve real-time control; in particular, the open-loop context is more numerically demanding than the closed-loop one. Significance. Our main contribution is the online feedback control of a noisy neuron through modulation of the input, taking into account physiological constraints on the control. A precise and robust targeting of neural activity based on stochastic optimal control has great potential for regulating neural activity in e.g. prosthetic applications and to improve our understanding of the basic mechanisms by which neuronal firing patterns can be controlled in vivo.
Highlights
We pose a stochastic optimal control problem to precisely specify the spike times in a leaky integrate-and-fire (LIF) model of a neuron with noise assumed to be of intrinsic or synaptic origin
Simulations show that our algorithms produce the desired results for the LIF model, and for the case where the neuron dynamics are given by more complex models than the LIF model
This is illustrated explicitly using the Morris–Lecar spiking neuron model, for which an LIF approximation is first obtained from a spike sequence using a previously published method
Summary
Namely targeting exact spike times in single neurons, has been considered mainly in the open-loop context, and in either absence of or for small noise. In [1] they use the spike response model, [15], to control output target spike trains and implement their scheme on pyramidal cells in mouse cortical slices Their method is numerically efficient and allows for the simultaneous control of many neurons. Our objective of imposing a certain timing sequence for the spike train using an externally applied control is obtained in both the closed- and open-loop settings, and we include the noise in the calculations of the controls. We compare the two methods through simulations against a simple-minded control technique which ignores the stochastic input to the neuron
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