Skip to main content

Advertisement

SpringerLink
Log in

Switching neuronal state: optimal stimuli revealed using a stochastically-seeded gradient algorithm

  • Published:
Journal of Computational Neuroscience Aims and scope Submit manuscript

Abstract

Inducing a switch in neuronal state using energy optimal stimuli is relevant to a variety of problems in neuroscience. Analytical techniques from optimal control theory can identify such stimuli; however, solutions to the optimization problem using indirect variational approaches can be elusive in models that describe neuronal behavior. Here we develop and apply a direct gradient-based optimization algorithm to find stimulus waveforms that elicit a change in neuronal state while minimizing energy usage. We analyze standard models of neuronal behavior, the Hodgkin-Huxley and FitzHugh-Nagumo models, to show that the gradient-based algorithm: 1) enables automated exploration of a wide solution space, using stochastically generated initial waveforms that converge to multiple locally optimal solutions; and 2) finds optimal stimulus waveforms that achieve a physiological outcome condition, without a priori knowledge of the optimal terminal condition of all state variables. Analysis of biological systems using stochastically-seeded gradient methods can reveal salient dynamical mechanisms underlying the optimal control of system behavior. The gradient algorithm may also have practical applications in future work, for example, finding energy optimal waveforms for therapeutic neural stimulation that minimizes power usage and diminishes off-target effects and damage to neighboring tissue.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

  • Aghababa, M. P., Amrollahi, M. H., & Borjkhani, M. (2012). Application of GA, PSO, and ACO algorithms to path planning of autonomous underwater vehicles. Journal of Marine Science and Application, 11(3), 378–386.

    Article  Google Scholar 

  • Aliev, R. R., & Panfilov, A. V. (1996). A simple two-variable model of cardiac excitation. Chaos, Solitons & Fractals, 7(3), 293–301.

    Article  Google Scholar 

  • Alle, H., Roth, A., & Geiger, J. R. P. (2009). Energy-efficient action potentials in hippocampal mossy fibers. Science, 325, 1405–1408.

    Article  CAS  PubMed  Google Scholar 

  • Alon, U. (2006). An introduction to systems biology - design principles of biological circuits. Boca Raton: CRC Press, Taylor & Francis Group, LLC.

    Google Scholar 

  • Attwell, D., & Laughlin, S. B. (2001). An energy budget for signaling in the grey matter of the brain. Journal of Cerebral Blood Flow and Metabolism, 21(10), 1133–45.

    Article  CAS  PubMed  Google Scholar 

  • Barnett, W., O’Brien, G., & Cymbalyuk, G. (2013). Bistability of silence and seizure-like bursting. Journal of Neuroscience Methods, 220(2), 179–89.

    Article  PubMed  Google Scholar 

  • Betts, J. J. T. (1998). Survey of numerical methods for trajectory optimization. Journal of Guidance, Control, and Dynamics, 21(2), 193–207.

    Article  Google Scholar 

  • Bryson, A. E., & Ho, Y.-C. (1975). Applied Optimal Control (Revised Pr.). Hemisphere Publishing Corporation.

  • Butera, R. J., Rinzel, J., & Smith, J. C. (1999). Models of respiratory rhythm generation in the pre-Bötzinger complex. I. Bursting pacemaker neurons. Journal of Neurophysiology, 82(1), 382–97.

    PubMed  Google Scholar 

  • Chen, Y., Wang, J., Wei, X., Deng, B., & Che, Y. (2011). Particle swarm optimization of periodic deep brain stimulation waveforms. Proceedings of the 30th Chinese Control Conference, 754–757.

  • Clay, J. R., Paydarfar, D., & Forger, D. B. (2008). A simple modification of the Hodgkin and Huxley equations explains type 3 excitability in squid giant axons. Journal of the Royal Society Interface, 5, 1421–1428.

    Article  PubMed Central  Google Scholar 

  • Clay, J. R., Forger, D., & Paydarfar, D. (2012). Ionic mechanism underlying optimal stimuli for neuronal excitation: role of Na + channel inactivation. PLoS One, 7(9), e45983.

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  • Danzl, P., Nabi, A., & Moehlis, J. (2010). Charge-balanced spike timing control for phase models of spiking neurons. Discrete and Continuous Dynamical Systems, 28(4), 1413–1435.

    Article  Google Scholar 

  • Dean, D. A., Forger, D. B., & Klerman, E. B. (2009). Taking the lag out of jet lag through model-based schedule design. PLoS Computational Biology, 5(6), e1000418.

    Article  PubMed Central  PubMed  Google Scholar 

  • Doležal, J. (1978). A gradient-type algorithm for the numerical solution of two-player zero-sum differential game problems. Kybernetika, 14(6), 429–446.

    Google Scholar 

  • Durand, D. M., & Warman, E. N. (1994). Desynchronization of epileptiform activity by extracellular current pulses in rat hippocampal slices. Journal of Physiology, 480(3), 527–537.

    CAS  PubMed Central  PubMed  Google Scholar 

  • Feng, X., Greenwald, B., & Rabitz, H. (2007). Toward closed-loop optimization of deep brain stimulation for Parkinson’s disease: concepts and lessons from a computational model. Journal of Neural Engineering, 4(2), L14–L21.

    Article  PubMed  Google Scholar 

  • FitzHugh, R. (1961). Impulses and physiological states in theoretical models of nerve membrane. Biophysical Journal, 1, 445–466.

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  • Forger, D. B., & Paydarfar, D. (2004). Starting, stopping, and resetting biological oscillators: in search of optimum perturbations. Journal of Theoretical Biology, 230, 521–532.

    Article  PubMed  Google Scholar 

  • Forger, D. B., Paydarfar, D., & Clay, J. R. (2011). Optimal stimulus shapes for neuronal excitation. PLoS Computational Biology, 7(7), e1002089.

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  • Gelfand IM & Fomin SV (2000). Calculus of Variations. Courier Dover Publications

  • Glass, L. (2001). Synchronization and rhythmic processes in physiology. Nature, 410(6825), 277–284.

    Article  CAS  PubMed  Google Scholar 

  • Golfetto, W. A., Fernandes, S., & da, S. (2012). A review of gradient algorithms for numerical computation of optimal trajectories. Journal of Aerospace Technology and Management, 4(2), 131–143.

    Article  Google Scholar 

  • Gupta, N., & Rink, R. (1973). Optimum control of epidemics. Mathematical Biosciences, 18, 383–396.

    Article  Google Scholar 

  • Hauptmann, C., Popovych, O., & Tass, P. A. (2005). Effectively desynchronizing deep brain stimulation based on a coordinated delayed feedback stimulation via several sites: a computational study. Biological Cybernetics, 93(6), 463–470.

    Article  CAS  PubMed  Google Scholar 

  • Hodgkin, A. L., & Huxley, A. F. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. Journal of Physiology, 117(4), 500–544.

    CAS  PubMed Central  PubMed  Google Scholar 

  • Howalski, C. H., da Silva, G. A., Poppi, R. J., Godoy, H. T., & Augusto, F. (2007). Neuro-genetic multioptimization of the determination of polychlorinated biphenyl congeners in human milk by headspace solid phase microextraction coupled to gas chromatography with electron capture detection. Analytica Chimica Acta, 585, 66–75.

    Article  Google Scholar 

  • Iasemidis, L. D. (2003). Epileptic seizure prediction and control. IEEE Transactions on Biomedical Engineering, 50(5), 549–558.

    Article  PubMed  Google Scholar 

  • Izhikevich, E. M. (2000). Neural excitability, spiking and bursting. International Journal of Bifurcation and Chaos, 10(6), 1171–1266.

    Article  Google Scholar 

  • Izhikevich EM (2007). Dynamical systems in neuroscience: the geometry of excitability and bursting. (TJ Sejnowski & TA Poggio, Eds.). Cambridge, Massachusetts: MIT Press

  • Jezernik, S., & Morari, M. (2005). Energy-optimal electrical excitation of nerve fibers. IEEE Transactions on Biomedical Engineering, 52(4), 740–743.

    Article  PubMed  Google Scholar 

  • Joshi, H. R. (2002). Optimal control of an HIV immunology model. Optimal Control Applications and Methods, 23(4), 199–213.

    Article  Google Scholar 

  • Kawato, M., & Suzuki, R. (1980). Two coupled neural oscillators as a model of the circadian pacemaker. Journal of Theoretical Biology, 86(3), 547–575.

    Article  CAS  PubMed  Google Scholar 

  • Kelley, H. J. (1962). Methods of Gradients. In G. Leitmann (Ed.), Optimization Techniques (5th ed., pp. 206–254). New York: Academic Press, Inc.

    Google Scholar 

  • Kepler, T. B., & Perelson, a. S. (1993). Somatic hypermutation in B cells: an optimal control treatment. Journal of Theoretical Biology, 164(1), 37–64.

    Article  CAS  PubMed  Google Scholar 

  • Kirschner, D., Lenhart, S., & Serbin, S. (1997). Optimal control of the chemotherapy of HIV. Journal of Mathematical Biology, 35(7), 775–792.

    Article  CAS  PubMed  Google Scholar 

  • Koelling, M. E., & Nykamp, D. Q. (2012). Searching for optimal stimuli: ascending a neuron’s response function. Journal of Computational Neuroscience, 33(3), 449–473.

    Article  PubMed  Google Scholar 

  • Lee, E. S. (1964). Optimization by a gradient technique. Industrial & Engineering Chemistry Fundamentals, 3(4), 373–380.

    Article  CAS  Google Scholar 

  • Lian, J., Bikson, M., Sciortino, C., Stacey, W. C., & Durand, D. M. (2003). Local suppression of epileptiform activity by electrical stimulation in rat hippocampus in vitro. Journal of Physiology, 547(2), 427–434.

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  • Loddenkemper, T., & Pan, A. (2001). Deep brain stimulation in epilepsy. Journal of Clinical Neurophysiology, 116(6), 217–34.

    Google Scholar 

  • Lozano, A. M. (2010). Deep brain stimulation for Parkinson’s disease. Journal of Neurosurgery, 112(3), 199–203.

    Article  Google Scholar 

  • Machens, C. K., Gollisch, T., Kolesnikova, O., & Herz, A. V. M. (2005). Testing the efficiency of sensory coding with optimal stimulus ensembles. Neuron, 47(3), 447–456.

    Article  CAS  PubMed  Google Scholar 

  • Moehlis, J., Shea-Brown, E., & Rabitz, H. (2006). Optimal inputs for phase models of spiking neurons. Journal of Computational and Nonlinear Dynamics, 1(4), 358–367.

    Article  Google Scholar 

  • Nabi, A., & Moehlis, J. (2012). Time optimal control of spiking neurons. Journal of Mathematical Biology, 64(6), 981–1004.

    Article  PubMed  Google Scholar 

  • Nagumo, J., Arimoto, S., & Yoshizawa, S. (1962). An active pulse transmission line simulating nerve axon. Proceedings of the IRE, 2061–2070.

  • Offner, F. (1946). Stimulation with minimum power. Journal of Neurophysiology, 9(5), 387–390.

    CAS  PubMed  Google Scholar 

  • Osborne, M. R. (1969). On shooting methods for boundary value problems. Journal of Mathematical Analysis and Applications, 27, 417–433.

    Article  Google Scholar 

  • Paydarfar, D., & Buerkel, D. D. (1995). Dysrhythmias of the respiratory oscillator. Chaos, 5(1), 18–29.

    Article  PubMed  Google Scholar 

  • Paydarfar, D., Forger, D. B., & Clay, J. R. (2006). Noisy inputs and the induction of on-off switching behavior in a neuronal pacemaker. Journal of Neurophysiology, 96(6), 3338–48.

    Article  PubMed  Google Scholar 

  • Raivo, T. (2000). Computational Methods for Dynamic Optimization and Pursuit-Evasion Games. Helsinki University of Technology.

  • Rao, A. (2010). A survey of numerical methods for optimal control. Advances in the Astronautical Sciences, 135, 497–528.

    Google Scholar 

  • Rinzel, J., & Ermentrout, G. B. (1998). Analysis of neural excitability and oscillations. In Methods in Neuronal Modeling (2nd ed., pp. 251–292).

  • Rubin, J. E., & Terman, D. (2004). High frequency stimulation of the subthalamic nucleus eliminates pathological thalamic rhythmicity in a computational model. Journal of Computational Neuroscience, 16(3), 211–235.

    Article  PubMed  Google Scholar 

  • Schiff, S. J. (2010). Towards model-based control of Parkinson’s disease. Philosophical Transactions of the Royal Society A, 368, 2269–2308.

    Article  Google Scholar 

  • Sengupta, B., Stemmler, M., Laughlin, S. B., & Niven, J. E. (2010). Action potential energy efficiency varies among neuron types in vertebrates and invertebrates. PLoS Computational Biology, 6(7), e1000840.

    Article  PubMed Central  PubMed  Google Scholar 

  • Serkh, K., & Forger, D. B. (2014). Optimal schedules of light exposure for rapidly correcting circadian misalignment. PLoS Computational Biology, 10(4), e1003523.

    Article  PubMed Central  PubMed  Google Scholar 

  • Sunderam, S., Gluckman, B., Reato, D., & Bikson, M. (2010). Toward rational design of electrical stimulation strategies for epilepsy control. Epilepsy & Behavior, 17(1), 6–22.

    Article  Google Scholar 

  • Tahayori, B., & Dokos, S. (2012). Optimal stimulus current waveshape for a Hodgkin-Huxley model neuron. 34th Annual International Conference of the IEEE EBS, 4627–4630.

  • Tass, P. A. (2003). A model of desynchronizing deep brain stimulation with a demand-controlled coordinated reset of neural subpopulations. Biological Cybernetics, 89(2), 81–88.

    Article  PubMed  Google Scholar 

  • Torrealdea, F. F., D’Anjou, A., Graña, M., Sarasola, C., & D’Anjou, A. (2006). Energy aspects of the synchronization of model neurons. Physical Review E, 74(1), 011905.

    Article  CAS  Google Scholar 

  • Watson, A., Barlow, H., & Robson, J. (1983). What does the eye see best? Nature, 302(5907), 419–422.

    Article  CAS  PubMed  Google Scholar 

  • Winfree, A. T. (2001). The Geometry of Biological Time. (J. E. Marsden, L. Sirovich, & S. Wiggins, Eds.) (2nd ed.). New York: Springer.

    Google Scholar 

  • Ypma, T. J. (1995). Historical development of the Newton–Raphson method. SIAM Review, 37(4), 531–551.

    Article  Google Scholar 

Download references

Acknowledgments

We thank Daniel Forger and Kirill Serkh for discussions on boundary value problems and also introducing us to the gradient algorithm. We thank John Clay for discussions regarding the ionic basis for excitability in squid axons, and Premananda Indic for discussions and guidance regarding gradient analysis. We also thank William Schwartz and anonymous reviewers for their suggestions and feedback on our manuscript. This work was supported by NIH R01 GM104987 and the Wyss Institute of Biologically Inspired Engineering. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Conflict of interest

The authors declare that they have no conflict of interest.

Author information

Authors and Affiliations

  1. Department of Neurology, University of Massachusetts Medical School, Worcester, MA, USA

    Joshua Chang & David Paydarfar

  2. Wyss Institute for Biologically Inspired Engineering, Harvard University, Boston, MA, USA

    David Paydarfar

Authors
  1. Joshua Chang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. David Paydarfar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Joshua Chang.

Additional information

Action Editor: J. Rinzel

Electronic supplementary material

Below is the link to the electronic supplementary material.

OR 1

Mathematical foundations of the gradient algorithm. A systematic tutorial in the mathematics underlying the gradient algorithm that we present in this paper. (PDF 90 kb)

OR 2

Implementation of gradient algorithm for the Hodgkin-Huxley and the Fitzhugh-Nagumo models. We show the matrices and equations used in the application of the gradient algorithm to the two models in this paper. (PDF 49 kb)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chang, J., Paydarfar, D. Switching neuronal state: optimal stimuli revealed using a stochastically-seeded gradient algorithm. J Comput Neurosci 37, 569–582 (2014). https://doi.org/10.1007/s10827-014-0525-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10827-014-0525-5

Keywords

Search

Navigation