Switching neuronal state: optimal stimuli revealed using a stochastically-seeded gradient algorithm

Abstract

Inducing a switch in neuronal state using energy optimal stimuli is relevant to a variety of problems in neuroscience. Analytical techniques from optimal control theory can identify such stimuli; however, solutions to the optimization problem using indirect variational approaches can be elusive in models that describe neuronal behavior. Here we develop and apply a direct gradient-based optimization algorithm to find stimulus waveforms that elicit a change in neuronal state while minimizing energy usage. We analyze standard models of neuronal behavior, the Hodgkin-Huxley and FitzHugh-Nagumo models, to show that the gradient-based algorithm: 1) enables automated exploration of a wide solution space, using stochastically generated initial waveforms that converge to multiple locally optimal solutions; and 2) finds optimal stimulus waveforms that achieve a physiological outcome condition, without a priori knowledge of the optimal terminal condition of all state variables. Analysis of biological systems using stochastically-seeded gradient methods can reveal salient dynamical mechanisms underlying the optimal control of system behavior. The gradient algorithm may also have practical applications in future work, for example, finding energy optimal waveforms for therapeutic neural stimulation that minimizes power usage and diminishes off-target effects and damage to neighboring tissue.

Evolution of extrema features reveals optimal stimuli for biological state transitions

“…Calculus of variations 50 is a well-established analytical framework used to address optimization problems. We solved the variational problem directly using a stochastically-seeded gradient algorithm 27 , applied to three models (defined in Supplementary Text S1 ) representing different types of transitions: from quiescence to a spike in the Hodgkin-Huxley model; between sustained oscillation and quiescence in the FitzHugh-Nagumo model; and between two steady states in a genetic toggle switch model. The stochastically-seeded gradient algorithm generates a series of random stimuli, and using a gradient-based approach, calculates how the system will respond to minute changes in the stimulus.…”

“…Analytical methods for deriving optimal control signals traditionally use calculus of variations 24 – 26 ; broad application of these techniques to biology is limited because validated mathematical models are available in only a few systems, and variational solutions to complex models can be elusive 27 , 28 . To deal with complex models, researchers have used phase reduction models 28 – 32 , parameterized stimuli 33 , or simpler conceptual models 24 , 25 .…”

Optimal stimulus waveforms for eliciting a spike: How close is the spike-triggered average?

“…Note that the shorter stimuli have higher amplitude and are right shifted, compared to the longer duration stimuli. We previously showed that the longer stimuli are more energetically optimal [15]. Because of this phenomenon, we can see that if longer and shorter stimuli are included in the library of snippets that induce a spike, averaging the briefer waveforms with the more prolonged waveforms would result in a right-shift compared to the global optimal calculated from the gradient algorithm.…”

“…This approach has been used to calculate energyoptimal stimuli for the HH model [8]. An alternative approach has solved this problem directly using a stochastically-seeded gradient algorithm [15]. We use this approach for the HH model, to determine the optimal stimulus necessary to trigger a single action potential in both high and low persistent current conditions.…”

“…Stochastic seeding enables the algorithm to explore a large solution space. In the case of the standard HH model (with no persistent current), we previously showed convergence to a global optimal waveform [15]. …”