Design of charge-balanced time-optimal stimuli for spiking neuron oscillators

Intensity-Varied Closed-Loop Noise Stimulation for Oscillation Suppression in the Parkinsonian State

This work explores the effectiveness of the intensity-varied closed-loop noise stimulation on the oscillation suppression in the Parkinsonian state. Deep brain stimulation (DBS) is the standard therapy for Parkinson's disease (PD), but its effects need to be improved. The noise stimulation has compelling results in alleviating the PD state. However, in the open-loop control scheme, the noise stimulation parameters cannot be self-adjusted to adapt to the amplitude of the synchronized neuronal activities in real time. Thus, based on the delayed-feedback control algorithm, an intensity-varied closed-loop noise stimulation strategy is proposed. Based on a computational model of the basal ganglia (BG) that can present the intrinsic properties of the BG neurons and their interactions with the thalamic neurons, the proposed stimulation strategy is tested. Simulation results show that the noise stimulation suppresses the pathological beta (12-35 Hz) oscillations without any new rhythms in other bands compared with traditional high-frequency DBS. The intensity-varied closed-loop noise stimulation has a more profound role in removing the pathological beta oscillations and improving the thalamic reliability than open-loop noise stimulation, especially for different PD states. And the closed-loop noise stimulation enlarges the parameter space of the delayed-feedback control algorithm due to the randomness of noise signals. We also provide a theoretical analysis of the effective parameter domain of the delayed-feedback control algorithm by simplifying the BG model to an oscillator model. This exploration may guide a new approach to treating PD by optimizing the noise-induced improvement of the BG dysfunction.

Nonlinear optimal control of a mean-field model of neural population dynamics

We apply the framework of nonlinear optimal control to a biophysically realistic neural mass model, which consists of two mutually coupled populations of deterministic excitatory and inhibitory neurons. External control signals are realized by time-dependent inputs to both populations. Optimality is defined by two alternative cost functions that trade the deviation of the controlled variable from its target value against the “strength” of the control, which is quantified by the integrated 1- and 2-norms of the control signal. We focus on a bistable region in state space where one low- (“down state”) and one high-activity (“up state”) stable fixed points coexist. With methods of nonlinear optimal control, we search for the most cost-efficient control function to switch between both activity states. For a broad range of parameters, we find that cost-efficient control strategies consist of a pulse of finite duration to push the state variables only minimally into the basin of attraction of the target state. This strategy only breaks down once we impose time constraints that force the system to switch on a time scale comparable to the duration of the control pulse. Penalizing control strength via the integrated 1-norm (2-norm) yields control inputs targeting one or both populations. However, whether control inputs to the excitatory or the inhibitory population dominate, depends on the location in state space relative to the bifurcation lines. Our study highlights the applicability of nonlinear optimal control to understand neuronal processing under constraints better.

Delayed Feedback-Based Suppression of Pathological Oscillations in a Neural Mass Model

Suppression of excessively synchronous beta frequency (12-35 Hz) oscillatory activity in the basal ganglia is believed to correlate with the alleviation of hypokinetic motor symptoms of the Parkinson's disease. Delayed feedback is an effective strategy to interrupt the synchronization and has been used in the design of closed-loop neuromodulation methods computationally. Although tremendous efforts in this are being made by optimizing delayed feedback algorithm and stimulation waveforms, there are still remaining problems in the selection of effective parameters in the delayed feedback control schemes. In most delayed feedback neuromodulation strategies, the stimulation signal is obtained from the local field potential (LFP) of the excitatory subthalamic nucleus (STN) neurons and then is administered back to STN itself only. The inhibitory external globus pallidus (GPe) nucleus in the excitatory-inhibitory STN-GPe reciprocal network has not been involved in the design of the delayed feedback control strategies. Thus, considering the role of GPe, this paper proposes three schemes involving GPe in the design of the delayed feedback strategies and compared their effectiveness to the traditional paradigm using STN only. Based on a neural mass model of STN-GPe network having capability of simulating the LFP directly, the proposed stimulation strategies are tested and compared. Our simulation results show that the four types of delayed feedback control schemes are all effective, even if with a simple linear delayed feedback algorithm. But the three new control strategies we propose here further improve the control performance by enlarging the oscillatory suppression space and reducing the energy expenditure, suggesting that they may be more effective in applications. This paper may guide a new approach to optimize the closed-loop deep brain stimulation treatment to alleviate the Parkinsonian state by retargeting the measurement and stimulation nucleus.

Entrainment of a network of interacting neurons with minimum stimulating charge

Periodic pulse train stimulation is generically used to study the function of the nervous system and to counteract disease-related neuronal activity, e.g., collective periodic neuronal oscillations. The efficient control of neuronal dynamics without compromising brain tissue is key to research and clinical purposes. We here adapt the minimum charge control theory, recently developed for a single neuron, to a network of interacting neurons exhibiting collective periodic oscillations. We present a general expression for the optimal waveform, which provides an entrainment of a neural network to the stimulation frequency with a minimum absolute value of the stimulating current. As in the case of a single neuron, the optimal waveform is of bang-off-bang type, but its parameters are now determined by the parameters of the effective phase response curve of the entire network, rather than of a single neuron. The theoretical results are confirmed by three specific examples: two small-scale networks of FitzHugh-Nagumo neurons with synaptic and electric couplings, as well as a large-scale network of synaptically coupled quadratic integrate-and-fire neurons.

The Geometry of Plasticity-Induced Sensitization in Isoinhibitory Rate Motifs

A well-known phenomenon in sensory perception is desensitization, wherein behavioral responses to persistent stimuli become attenuated over time. In this letter, our focus is on studying mechanisms through which desensitization may be mediated at the network level and, specifically, how sensitivity changes arise as a function of long-term plasticity. Our principal object of study is a generic isoinhibitory motif: a small excitatory-inhibitory network with recurrent inhibition. Such a motif is of interest due to its overrepresentation in laminar sensory network architectures. Here, we introduce a sensitivity analysis derived from control theory in which we characterize the fixed-energy reachable set of the motif. This set describes the regions of the phase-space that are more easily (in terms of stimulus energy) accessed, thus providing a holistic assessment of sensitivity. We specifically focus on how the geometry of this set changes due to repetitive application of a persistent stimulus. We find that for certain motif dynamics, this geometry contracts along the stimulus orientation while expanding in orthogonal directions. In other words, the motif not only desensitizes to the persistent input, but heightens its responsiveness (sensitizes) to those that are orthogonal. We develop a perturbation analysis that links this sensitization to both plasticity-induced changes in synaptic weights and the intrinsic dynamics of the network, highlighting that the effect is not purely due to weight-dependent disinhibition. Instead, this effect depends on the relative neuronal time constants and the consequent stimulus-induced drift that arises in the motif phase-space. For tightly distributed (but random) parameter ranges, sensitization is quite generic and manifests in larger recurrent E-I networks within which the motif is embedded.