Optimal Inputs for Phase Models of Spiking Neurons

Variational methods are used to determine the optimal currents that elicit spikes in various phase reductions of neural oscillator models. We show that, for a given reduced neuron model and target spike time, there is a unique current that minimizes a squareintegral measure of its amplitude. For intrinsically oscillatory models, we further demonstrate that the form and scaling of this current is determined by the model’s phase response curve. These results reflect the role of intrinsic neural dynamics in determining the time course of synaptic inputs to which a neuron is optimally tuned to respond, and are illustrated using phase reductions of neural models valid near typical bifurcations to periodic firing, as well as the Hodgkin-Huxley equations. DOI: 10.1115/1.2338654 Phase-reduced models of neurons have traditionally been used to investigate either the patterns of synchrony that result from the type and architecture of coupling 1–8 or the response of large groups of oscillators to external stimuli 9–11. In all of these cases, the inputs to the model cells were fixed by definition of the model at the outset and the dynamics of phase models of networks or populations were analyzed in detail. The present paper takes a complementary, control-theoretic approach that is related to probabilistic “spike-triggered” methods 12: we fix at the outset a feature of the dynamical trajectories of interest—spiking at a precise time t1—and study the neural inputs that lead to this outcome. By computing the optimal such input, according to a measure of the input strength required to elicit the spike, we identify the signal to which the neuron is optimally “tuned” to respond. We view the present work as part of the first attempts 13,14 to understand the dynamical response of neurons using control theory, and, as we expect that insights from this general perspective will be combined with the “forward” dynamics results that Phil Holmes and many others have derived to ultimately enhance our understanding of neural processing, we hope that it will serve as a fitting tribute to his work.

[1]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[2]  A. Hodgkin,et al.  A quantitative description of membrane current and its application to conduction and excitation in nerve , 1952, The Journal of physiology.

[3]  H. C. Corben,et al.  Classical Mechanics (2nd ed.) , 1961 .

[4]  S. Kahne,et al.  Optimal control: An introduction to the theory and ITs applications , 1967, IEEE Transactions on Automatic Control.

[5]  Arthur E. Bryson,et al.  Applied Optimal Control , 1969 .

[6]  A. Winfree.  The geometry of biological time , 1991 .

[7]  J. W. Humberston.  Classical mechanics , 1980, Nature.

[8]  G. Kraepelin,et al.  A. T. Winfree, The Geometry of Biological Time (Biomathematics, Vol.8). 530 S., 290 Abb. Berlin‐Heidelberg‐New‐York 1980. Springer‐Verlag. DM 59,50 , 1981 .

[9]  P. Holmes,et al.  The nature of the coupling between segmental oscillators of the lamprey spinal generator for locomotion: A mathematical model , 1982, Journal of mathematical biology.

[10]  G. Ermentrout,et al.  Frequency Plateaus in a Chain of Weakly Coupled Oscillators, I. , 1984 .

[11]  G. Ermentrout,et al.  Phase transition and other phenomena in chains of coupled oscilators , 1990 .

[12]  P. Ashwin,et al.  The dynamics ofn weakly coupled identical oscillators , 1992 .

[13]  D. Hansel,et al.  Phase Dynamics for Weakly Coupled Hodgkin-Huxley Neurons , 1993 .

[14]  Wulfram Gerstner,et al.  What Matters in Neuronal Locking? , 1996, Neural Computation.

[15]  William Bialek,et al.  Spikes: Exploring the Neural Code , 1996 .

[16]  Bard Ermentrout,et al.  Type I Membranes, Phase Resetting Curves, and Synchrony , 1996, Neural Computation.

[17]  Yoshiki Kuramoto,et al.  Phase- and Center-Manifold Reductions for Large Populations of Coupled Oscillators with Application to Non-Locally Coupled Systems , 1997 .

[18]  P. Holmes,et al.  Simple models for excitable and oscillatory neural networks , 1998, Journal of mathematical biology.

[19]  P. Tass.  Phase Resetting in Medicine and Biology , 1999 .

[20]  G. Estabrook.  The Geometry of Biological Time.Second Edition. Interdisciplinary Applied Mathematics, Volume 12. ByArthur T Winfree. New York: Springer. $89.95. xxvi + 777 p; ill.; index of author citations and publications, index of subjects. ISBN: 0–387–98992–7. 2001. , 2002 .

[21]  Bard Ermentrout,et al.  Simulating, analyzing, and animating dynamical systems - a guide to XPPAUT for researchers and students , 2002, Software, environments, tools.

[22]  P. Holmes,et al.  Globally Coupled Oscillator Networks , 2003 .

[23]  Jianfeng Feng,et al.  Optimal control of neuronal activity. , 2003, Physical review letters.

[24]  Adrienne L. Fairhall,et al.  Computation in a Single Neuron: Hodgkin and Huxley Revisited , 2002, Neural Computation.

[25]  Liaofu Luo,et al.  Minimal model for genome evolution and growth. , 2002, Physical review letters.

[26]  Philip Holmes,et al.  A Minimal Model of a Central Pattern Generator and Motoneurons for Insect Locomotion , 2004, SIAM J. Appl. Dyn. Syst..

[27]  Philip Holmes,et al.  The Influence of Spike Rate and Stimulus Duration on Noradrenergic Neurons , 2004, Journal of Computational Neuroscience.

[28]  David Paydarfar,et al.  Starting, stopping, and resetting biological oscillators: in search of optimum perturbations. , 2004, Journal of theoretical biology.

[29]  Wulfram Gerstner,et al.  Noise and the PSTH Response to Current Transients: I. General Theory and Application to the Integrate-and-Fire Neuron , 2001, Journal of Computational Neuroscience.

[30]  Eric Shea-Brown,et al.  On the Phase Reduction and Response Dynamics of Neural Oscillator Populations , 2004, Neural Computation.

[31]  A. Rothman,et al.  Exploring the level sets of quantum control landscapes (9 pages) , 2006 .

[32]  Jeff Moehlis,et al.  Event-based feedback control of nonlinear oscillators using phase response curves , 2007, 2007 46th IEEE Conference on Decision and Control.

[33]  P. Danzl,et al.  Spike timing control of oscillatory neuron models using impulsive and quasi-impulsive charge-balanced inputs , 2008, 2008 American Control Conference.