L. Li, I.M. Park, A. Brockmeier, B. Chen, S. Seth, J. T. Francis, J. C. Sanchez, and J. C. Príncipe
The precise control of spiking in a population of neurons via applied electrical stimulation is a challenge due to the sparseness of spiking responses and neural system plasticity. We pose neural stimulation as a system control problem where the system input is a multidimensional time-varying signal representing the stimulation, and the output is a set of spike trains; the goal is to drive the output such that the elicited population spiking activity is as close as possible to some desired activity, where closeness is defined by a cost function. If the neural system can be described by a time-invariant (homogeneous) model, then offline procedures can be used to derive the control procedure; however, for arbitrary neural systems this is not tractable. Furthermore, standard control methodologies are not suited to directly operate on spike trains that represent both the target and elicited system response. In this paper, we propose a multiple-input multiple-output (MIMO) adaptive inverse control scheme that operates on spike trains in a reproducing kernel Hilbert space (RKHS). The control scheme uses an inverse controller to approximate the inverse of the neural circuit. The proposed control system takes advantage of the precise timing of the neural events by using a Schoenberg kernel defined directly in the space of spike trains. The Schoenberg kernel maps the spike train to an RKHS and allows linear algorithm to control the nonlinear neural system without the danger of converging to local minima. During operation, the adaptation of the controller minimizes a difference defined in the spike train RKHS between the system and the target response and keeps the inverse controller close to the inverse of the current neural circuit, which enables adapting to neural perturbations. The results on a realistic synthetic neural circuit show that the inverse controller based on the Schoenberg kernel outperforms the decoding accuracy of other models based on the conventional rate- representation of neural signal (i.e., spikernel and generalized linear model). Moreover, after a significant perturbation of the neuron circuit, the control scheme can successfully drive the elicited responses close to the original target responses.