Optimal design of minimum-power stimuli for phase models of neuron oscillators

Abstract

In this paper, we study optimal control problems of spiking neurons whose dynamics are described by a phase model. We design minimum-power current stimuli (controls) that lead to targeted spiking times. In particular, we consider bounded control amplitude and characterize the range of possible spiking times determined by the bound, which can be chosen sufficiently small within the range where the phase model is valid. We show that for a given bound the corresponding feasible spiking times are optimally achieved by piecewise continuous controls. We present analytic expressions with numerical simulations of the minimum-power stimuli for several phase models. We demonstrate the applicability of our method with an experimentally determined phase response curve.

Design of Charge-Balanced Time-Optimal Stimuli for Spiking Neuron Oscillators

“…where α is a constant defined in (12). Since the minimum-time control takes the bangbang form as shown in Section 2.1.1, it requires to calculate the switching points and determine the type of the switching at these points for the optimal control synthesis.…”

“…It has been observed through experiments that the PRC for an Aplysia motoneuron is extremely similar to that of a Morris-Lecar PRC [29]. We consider a Morris-Lecar system with parameter values given in [12], which has a natural frequency ω = 0.283 rad/ms. The PRC is approximated by (26) with the coefficients shown in Table 2 and is illustrated, with its derivatives, in Figure 5 Table 2.…”

“…Recently, controltheoretic approaches, including calculus of variations and the maximum principle, have been employed to design external stimuli for optimal manipulation of the dynamic behavior of neuron oscillators. These include the design of minimum-power controls for spiking a single neuron at specified time instances [11,12,13], optimal waveforms for entrainment of neuron ensembles [14,15,16], and open-loop controls for establishing and maintaining a desired phase configuration, such as anti-phase for two coupled neuron oscillators [17]. Work on considering stochastic effects to neuron systems such as the optimal control of neuronal spiking activity receiving a class of random synaptic inputs has also been investigated [18].…”

Optimal entrainment with smooth, pulse, and square signals in weakly forced nonlinear oscillators

“…[8,9,17,25], optimal forcings have been designed for various purposes and constraints. Their arguments are all based on variational calculus, such as the Euler-Lagrange equation [26] or Pontryagin's minimum principle [27].…”

“…The constraints in Eq. (9) imply that there exists an upper bound of J[f ], i.e., the ideal locking range R[f ] under the constraints of (4) and (5). In addition, for the case of 1 < p < ∞ mentioned above, the equality condition has to be satisfied for this ideal locking range to be realized.…”