Abstract
We present an event-based feedback control method for randomizing the asymptotic phase of oscillatory neurons. Phase randomization is achieved by driving the neuron’s state to its phaseless set, a point at which its phase is undefined and is extremely sensitive to background noise. We consider the biologically relevant case of a fixed magnitude constraint on the stimulus signal, and show how the control objective can be accomplished in minimum time. The control synthesis problem is addressed using the minimum-time-optimal Hamilton–Jacobi–Bellman framework, which is quite general and can be applied to any spiking neuron model in the conductance-based Hodgkin–Huxley formalism. We also use this methodology to compute a feedback control protocol for optimal spike rate increase. This framework provides a straightforward means of visualizing isochrons, without actually calculating them in the traditional way. Finally, we present an extension of the phase randomizing control scheme that is applied at the population level, to a network of globally coupled neurons that are firing in synchrony. The applied control signal desynchronizes the population in a demand-controlled way.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Åström KJ, Bernhardsson B (2003) Systems with Lebesgue sampling. In: Rantzer A, Byrnes CI (eds) Directions in mathematical systems theory and optimization, vol XIII. Springer, Berlin
Athans M, Falb PL (1966) Optimal control: an introduction to the theory and its applications. McGraw-Hill, New York
Bardi M, Capuzzo-Dolcetta I (1997) Optimal control and viscosity solutions of Hamilton–Jacobi–Bellman equations. Birkhauser, Boston
Benabid AL, Pollak P, Gervason C, Hoffmann D, Gao DM, Hommel M, Perret JE, De Rougemont J (1991) Long-term suppression of tremor by chronic stimulation of the ventral intermediate thalamic nucleus. Lancet 337: 403–406
Campbell A, Gonzalez A, Gonzalez DL, Piro O, Larrondo HA (1989) Isochrones and the dynamics of kicked oscillators. Physica A 155(3): 565–584
Danzl P, Moehlis J (2007) Event-based feedback control of nonlinear oscillators using phase response curves. In: Proceedings of the 46th IEEE conference on decision and control, pp 5806–5811, New Orleans, LA
Danzl P, Moehlis J (2008) Spike timing control of oscillatory neuron models using impulsive and quasi-impulsive charge-balanced inputs. In: Proceedings of the 2008 American control conference, pp 171–176, Seattle, WA
Danzl P, Hansen R, Bonnet G, Moehlis J (2008) Partial phase synchronization of neural populations due to random Poisson inputs. J Comput Neurosci 25(1): 141–157
FitzHugh R (1961) Impulses and physiological states in theoretical models of nerve membrane. Biophys J 1(6): 445–466
Guckenheimer J (1975) Isochrons and phaseless sets. J Math Biol 1: 259–273
Hodgkin AL, Huxley AF (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol 117: 500–544
Josic K, Shea-Brown ET, Moehlis J (2006) Isochron. Scholarpedia 1(8):1361. http://www.scholarpedia.org
Keener J, Sneyd J (1998) Mathematical physiology. Springer, New York
Khalil HK (2002) Nonlinear systems. Prentice Hall, Upper Saddle River
Mitchell IM (2007) A toolbox of level set methods. Technical Report UBC CS TR-2007-11
Mitchell IM (2008) The flexible, extensible and efficient toolbox of level set methods. J Sci Comput 35(2): 300–329
Moehlis J (2006) Canards for a reduction of the Hodgkin–Huxley equations. J Math Biol 52(2): 141–153
Nagumo J, Arimoto S, Yoshizawa S (1962) An active pulse transmission line simulating nerve axon. Proc IRE 50(10): 2061–2070
Netoff TI, Acker CD, Bettencourt JC, White JA (2005) Beyond two-cell networks: experimental measurement of neuronal responses to multiple synaptic inputs. J Comput Neurosci 18(3): 287–295
Nini A, Feingold A, Slovin H, Bergman H (1995) Neurons in the globus pallidus do not show correlated activity in the normal monkey, but phase-locked oscillations appear in the MPTP model of Parkinsonism. J Neurophysiol 74(4): 1800–1805
Osher S (1993) A level set formulation for the solution of the Dirichlet problem for Hamilton–Jacobi equations. SIAM J Math Anal 24: 1145
Pare D, Curro’Dossi R, Steriade M (1990) Neuronal basis of the Parkinsonian resting tremor: a hypothesis and its implications for treatment. Neuroscience 35: 217–226
Pontryagin LS, Trirogoff KN, Neustadt LW (1962) The mathematical theory of optimal processes. Wiley, New York
Popovych OV, Hauptmann C, Tass PA (2006) Control of neuronal synchrony by nonlinear delayed feedback. Biol Cybern 95(1): 69–85
Steriade M (2003) Neuronal substrates of sleep and epilepsy. Cambridge University Press, New York
Tass PA (1999) Phase resetting in medicine and biology. Springer, New York
Tass PA (2000) Effective desynchronization by means of double-pulse phase resetting. Europhys Lett 53: 15–21
Winfree A (2001) The geometry of biological time, 2nd edn. Springer, New York
Acknowledgments
We thank G. Orosz, P. Atzberger, and E. Shea-Brown for insightful discussions regarding this study. We also recognize I. Mitchell for his exceptional Level Set Methods Toolbox, which provided several numerical routines that were important for generating these results. P. Danzl is supported by the National Science Foundation through the Integrative Graduate Education and Research Traineeship program. J. Hespanha is supported by the National Science Foundation under grates ECCS-0725485 and ECCS-0835847. J. Moehlis is supported by the National Science Foundation under Grant NSF-0547606.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Danzl, P., Hespanha, J. & Moehlis, J. Event-based minimum-time control of oscillatory neuron models. Biol Cybern 101, 387–399 (2009). https://doi.org/10.1007/s00422-009-0344-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00422-009-0344-3