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Research Papers

Current Stimuli That Provide Membrane Voltage Tracking in a Six Dimensional Neuron Model

[+] Author and Article Information
Melinda E. Koelling

Department of Mathematics,
Western Michigan University,
Kalamazoo, MI 49008
e-mail: melinda.koelling@wmich.edu

Damon A. Miller

e-mail: damon.miller@wmich.edu

Michael Ellinger

e-mail: michael.e.ellinger@wmich.edu

Frank L. Severance

e-mail: frank.severance@wmich.edu

John Stahl

e-mail: john.stahl@wmich.edu
Department of Electrical and Computer Engineering,
Western Michigan University,
Kalamazoo, MI 49008

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received January 30, 2012; final manuscript received January 23, 2013; published online May 22, 2013. Assoc. Editor: Sergey Nersesov.

J. Dyn. Sys., Meas., Control 135(4), 041014 (May 22, 2013) (9 pages) Paper No: DS-12-1040; doi: 10.1115/1.4023667 History: Received January 30, 2012; Revised January 23, 2013

Optimization techniques have been applied to neuron models for a variety of purposes, including control of neuron firing rates and minimizing input stimulus current magnitudes. Optimal control is used to minimize a quantity of interest; often, the time or energy needed to complete an objective. Rather than attempting to control or modify neuron dynamics, this paper demonstrates that optimal control can be used to obtain an optimal input stimulus current i*(t) which causes a six dimensional Hodgkin–Huxley type neuron model to approximate a specified reference membrane voltage. The reference voltages considered in this paper consist of one or more action potentials as evoked by an input current i(t). In the described method, the user prescribes a balance of low squared integral of input stimulus current (input stimulus “energy”) and accurate tracking of the original reference voltage. In a previous work, the authors applied this approach to a reduced order neuron model. This paper demonstrates the applicability of this technique to biologically plausible higher dimensional conductance based neuron models. For each investigated neuron response, the method discovered optimal input stimuli current i*(t) having a lower energy than the original i(t), while still providing accurate tracking of the reference voltage.

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References

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Figures

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Fig. 1

Function f(v) for the three cases considered in this paper. Zero crossings of f(v) correspond to values of the membrane voltage v at equilibrium. As in [8], filled and unfilled circles indicate stable and unstable equilibrium points, respectively. Note that the only difference between the double and multiple spike cases is the value of i(t), resulting in a vertical shift of f(v).

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Fig. 2

Simulation results for the single spike case. Left column: as α is increased, the energy of the input stimulus current i*(t) (solid line) is reduced, as compared to the current i(t) (dashed line) that evoked r(t). This is at the expense of an increasing tracking error, as noted by comparing r(t) (dashed line) to the membrane voltage v*(t) (solid line) associated with i*(t) in the right column.

Grahic Jump Location
Fig. 3

Simulation results for the double spike case. Left column: as α is increased, the energy of the input stimulus current i*(t) (solid line) is reduced, as compared to the current i(t) (dashed line) that evoked r(t). This at the expense of an increasing tracking error, as noted by comparing r(t) (dashed line) to the membrane voltage v*(t) (solid line) associated with i*(t) in the right column.

Grahic Jump Location
Fig. 4

Simulation results for the multiple spike case. Left column: as α is increased, the energy of the input stimulus current i*(t) (solid line) is reduced, as compared to the current i(t) (dashed line) that evoked r(t). This at the expense of an increasing tracking error, as noted by comparing r(t) (dashed line) to the membrane voltage v*(t) (solid line) associated with i*(t) in the right column.

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Fig. 5

A closer view of two spikes from the multiple spike case for α = 0.9

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Fig. 6

Results show a shift in the spike timing for the single spike case (left column) but not the double spike case (right column) as α is increased

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Fig. 7

The right column plots J[i*(t)] and its constituent terms, the α/2 weighted energy and the (1 − α)/2 weighted tracking error for three cases for various values of α. The left column is the same plot, except that the effect of α has been removed. In all cases, the energy decreases and the tracking error increases as α increases. For comparison, i(t) has an unweighted “energy” of 25, 4, and 900 for the single, double, and multiple spike cases, respectively. Note that some unweighted tracking error values for the single and double spike cases are outside the plot area for α ≈ 1.

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