Single input optimal control for globally coupled neuron networks

Abstract

We consider the problem of desynchronizing a network of synchronized, globally (all-to-all) coupled neurons using an input to a single neuron. This is done by applying the discrete time dynamic programming method to reduced phase models for neural populations. This technique numerically minimizes a certain cost function over the whole state space, and is applied to a Kuramoto model and a reduced phase model for Hodgkin-Huxley neurons with electrotonic coupling. We evaluate the effectiveness of control inputs obtained by averaging over results obtained for different coupling strengths. We also investigate the applicability of this method to Hodgkin-Huxley models driven by multiplicative stimuli.

Synchronization in complex networks of phase oscillators: A survey

“…Technological applications of the coupled oscillator model (1) and its generalization (4) include deep brain stimulation (Tass, 2003;Nabi and Moehlis, 2011;Franci et al, 2012), locking in solid-state circuit oscillators (Abidi and Chua, 1979;Mirzaei et al, 2007), planar vehicle coordination Sepulchre et al, 2007Sepulchre et al, , 2008Klein, 2008;Klein et al, 2008), carrier synchronization without phase-locked loops (Rahman et al, 2011), synchronization in semiconductor laser arrays (Kozyreff et al, 2000), and microwave oscillator arrays (York and Compton, 2002). Since alternating current (AC) circuits are naturally modeled by equations similar to (1), some electric applications are found in structure-preserving (Bergen and Hill, 1981;Sauer and Pai, 1998) and networkreduced power system models (Chiang et al, 1995;Dörfler and Bullo, 2012b), and droop-controlled inverters in microgrids (Simpson-Porco et al, 2013).…”

“…Applications in neuroscience (Crook et al, 1997;Varela et al, 2001;Brown et al, 2003), deep-brain stimulation (Tass, 2003;Nabi and Moehlis, 2011;Franci et al, 2012), vehicle coordination Sepulchre et al, 2007Sepulchre et al, , 2008Klein, 2008;Klein et al, 2008), and central pattern generators for locomotion purposes (Ijspeert, 2008;Aoi and Tsuchiya, 2005;Righetti and Ijspeert, 2006) motivate the study of coherent behaviors with synchronized frequencies where the phases are not cohesive, but rather dispersed in appropriate patterns. Whereas the phase-synchronized state is characterized by the order parameter r achieving its maximal (unit) magnitude, we say that a solution θ : R ≥0 → T n to the coupled oscillator model (1) achieves phase balancing if all phases θ i (t) asymptotically converge to the set…”

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

“…The inter-spike time interval of a neuron characterizes its properties and can be controlled by use of external stimuli. The ability to control neuron spiking activities is fundamental to theoretical neuroscience, and the concept of effective control of such neurological behavior has led to the development of innovative therapeutic procedures [2,3] for neurological disorders including deep brain stimulation (DBS) for Parkinson's disease and essential tremor [4,5], where electrical pulses are used to inhibit pathological synchrony among neuron populations. In such neurological treatments and other applications such as the design of artificial cardiac pacemakers [6], it is of clinical importance to avoid long and strong electrical pulses in order to prevent the tissue from damage, as well as to maintain zero net electric charge accumulation over each stimulation cycle in order to suppress undesirable side effects.…”

Minimum energy spike randomization for neurons

“…Once the solution V(z, t) is computed, the optimal control is found as a function of the state at all time steps (see (14)). Given this data in time and space, we set the initial condition for the system (2) to be the spiking point (V 0 , n 0 ) = (44.8, 0.459) ≡ (V s , n s ), and find the associated optimal control sequence u(t) through forward integration of (2).…”

“…These methods are attractive from a clinical perspective in that the control stimulus is applied only when needed (characterized by the feedback signal) and in an optimal way (characterized by the optimality criteria). There are examples of these both on a single neuron level [5], [6], [7], [8], [9] and on a population level [10], [11], [12], [13], [14]. Other studies have also shown potential in desynchronizing a population of pathologically synchronized neurons [15], [16], [17], [18], [19].…”