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RESEARCH PAPERS

Optimal Inputs for Phase Models of Spiking Neurons

[+] Author and Article Information
Jeff Moehlis

Department of Mechanical Engineering, University of California, Santa Barbara, Santa Barbara, CA 93106moehlis@engineering.ucsb.edu

Eric Shea-Brown

Courant Institute for Mathematical Sciences, and Center for Neural Science, New York University, New York, NY 10012ebrown@math.nyu.edu

Herschel Rabitz

Department of Chemistry, Princeton University, Princeton, NJ 08544hrabitz@princeton.edu

J. Comput. Nonlinear Dynam. 1(4), 358-367 (Jun 03, 2006) (10 pages) doi:10.1115/1.2338654 History: Received November 21, 2005; Revised June 03, 2006

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 square-integral 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.

Copyright © 2006 by American Society of Mechanical Engineers
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Figures

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Figure 2

Dependence of t1 on λ0 for the sinusoidal PRC 319, as obtained from 320

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Figure 15

Minimal time of firing tfbb as a function of I¯, obtained using bang-bang control, for phase models starting at θi=0 for (a) solid line: f(θ)=ω=1, Z(θ)=sinθ; dashed line: f(θ)=ω=1, Z(θ)=1−cosθ; dotted-dashed line: the theta neuron model with Ib=0.25; dotted line: the theta neuron model with Ib=−0.25, and (b) the PRC for the Hodgkin-Huxley equations with standard parameters and Ib=10

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Figure 1

Phase space for 24,25 with the sinusoidal PRC 319 and ω=Zd=1, showing fixed points at (θ,λ)=(π∕2,−2) and (3π∕2,−2), stable and unstable manifolds of the fixed points, and trajectories with t1=5 and t1=9

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Figure 5

Scaling of the amplitude of optimal currents with baseline frequency ω, for (a) the sinusoidal PRC Z(θ)=(1∕ω−ωH)sin(θ), with ωH=0.5 and (b) the SNIPER PRC Z(θ)=(1∕ω)[1−cos(θ)]. For t1−T=−0.5, the amplitude Imax1 from the lowest-order expression 217 is given by solid lines; stars give the analogous numerically computed values (i.e., to all orders). For the fraction p=0.9, the amplitude Imax2 from the lowest-order expression 218 is given by dotted-dashed lines; triangles give the analogous numerical values. Insets give the same data on log-log axes.

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Figure 6

Phase space for 24,25 for the SNIPER PRC 322 with ω=1 and Zd=1, showing the fixed point at (θ,λ)=(π,−1∕2), stable and unstable manifolds of the fixed point, and trajectories for periodic orbits with period t1=5 and t1=9

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Figure 7

Optimal currents for the SNIPER PRC 322 with ω=Zd=1 for different values of t1, with scaled time axis for ease of comparison

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Figure 8

Phase space for 24,25 for the theta neuron model 323 with (a) Ib=0.25, (b) Ib=−0.25, showing fixed points, stable and unstable manifolds of the fixed points, and trajectories for periodic orbits with period t1=5 and t1=9. The dot in (b) is a center fixed point.

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Figure 9

Optimal currents for the theta neuron model, (a) with Ib=0.25 and (b) with Ib=−0.25, with time axis scaled as above. Target time values are, from top, t1=3,4,5,6,10,25.

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Figure 10

Phase response curve for the Hodgkin-Huxley equations with standard parameters and injected baseline current IHH=10

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Figure 11

Phase space for 24,25 for the PRC corresponding to the Hodgkin-Huxley equations with IHH=10, showing the stable and unstable manifolds of the two fixed points, and trajectories for periodic orbits with period t1=14 and t1=18

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Figure 12

Optimal currents for the PRC for the Hodgkin-Huxley equations with standard parameters and with IHH=10 for different values of t1, with scaled time axis for ease of comparison

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Figure 13

Dynamics of the full Hodgkin-Huxley equations with I(t) chosen to be the optimal current stimulus for t1=14 for the phase model with the Hodgkin-Huxley PRC for IHH=10. (a) shows the time series of the transmembrane voltage V, and (b) shows the phase space projection onto the (V,n) plane, where V is the voltage and n is a gating variable (using the standard Hodgkin-Huxley notation). The thin line shows the dynamics while I(t) is being applied up to time t1. The thick line shows the dynamics after I(t) is turned off until the neuron first fires.

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Figure 14

Comparison of the specified time of firing t1 and the actual time of firing t1HH for the full Hodgkin-Huxley equations for the current found from optimizing the phase model. The dashed line corresponds to exact agreement.

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Figure 3

Optimal currents for the sinusoidal PRC 319 with ω=Zd=1 for different values of t1, with scaled time axis for ease of comparison

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Figure 4

Exact (solid lines) and approximate (dashed lines) optimal currents for t1 as labeled with (a) the sinusoidal PRC 319 with ω=Zd=1, (b) the SNIPER PRC 322 with ω=Zd=1, and (c) the PRC corresponding to the Hodgkin-Huxley equations with Ib=10

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