Fokker-Planck Solution for a Neuronal Spiking Model Derek J. Daniel a a

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1 This article was downloaded by: [Montana State University] On: 31 March 2010 Access details: Access Details: [subscription number ] Publisher Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Transport Theory and Statistical Physics Publication details, including instructions for authors and subscription information: Fokker-Planck Solution for a Neuronal Spiking Model Derek J. Daniel a a Department of Physical Science and Electrical, Electronic, Engineering, Faculty of Engineering and Science, University Tunku Abdul Rahman, Kuala Lumpur, Malaysia Online publication date: 30 November 2009 To cite this Article Daniel, Derek J.(2009) 'Fokker-Planck Solution for a Neuronal Spiking Model', Transport Theory and Statistical Physics, 38: 7, To link to this Article: DOI: / URL: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 Transport Theory and Statistical Physics, 38: , 2009 Copyright C Taylor & Francis Group, LLC ISSN: print / online DOI: / FOKKER-PLANCK SOLUTION FOR A NEURONAL SPIKING MODEL DEREK J. DANIEL University Tunku Abdul Rahman, Department of Physical Science and Electrical, Electronic, Engineering, Faculty of Engineering and Science, Kuala Lumpur, Malaysia The stochastic dynamics of a neuronal spiking model in neuroscience, when viewed as a large simulated network, are known to reduce to the classic problem of solving the Fokker-Planck equation, or the equivalent Kolmogorov differential equation in probability theory, for the numerical evaluation of the statistical properties of neurons as a random injection of ion currents. The problem here, however, is that the initial condition for the Fokker-Planck equation is a Dirac delta function so the actual implementation of Delta functions that at the same time can attain numerical stability can become problematic in computational neuroscience. Therefore, in this brief communication, a computational method for implementing such an initial condition is suggested, which itself has led to an exact solution for this problem. Keywords: Dirac Delta-Function Condition, Ion-Current Transport Phenomena, Kolmogorov Master Equation, Langevin Equation, Magnetic-Resonance, Neuron Spikes, Neuroscience, Postsynaptic and Resting Membrane Potentials, Stochastic Dynamics Neuron Stochastic-Spike Dynamics: A Brief Introduction Observed phenomena in the cortex, such as perception, speech, learning, and language, to name a few, are readily expressed as the collective dynamics of large populations of interacting neurons (Reutimann et al., 2003). Neurons, as readily understood and comprehensively reviewed in neuroscience (Reutimann et al., 2003; Decol et al., 2008; Gerstner and Kistler, 2002) and references therein), are the cells responsible for the migration Received 1 June, 2009, Accepted 30 September, 2009 Address correspondence to Derek J. Daniel, University Tunku Abdul Rahman, Department of Physical Science and Electrical, Electronic, Engineering, Faculty of Engineering & Science, Kuala Lumpur 53300, Malaysia. djdaniel@reloaded.aceshells.com 383

3 384 D. J. Daniel and encoding of signals within the nervous system. Following this understanding, one can also explain the neuron s transmission dynamics as a simulated network of exchanges of information among neurons, which is known to occur when the electrical potentials of the neurons are at rest, referenced as theresting membrane potential. Furthermore, postsynaptic potentials arise whenever the synapses of a neuron receives input from other neurons in such a way that induces transient behavior in the neuron s resting membrane potential. It is this transient behavior that one recognizes as being due to the flux of ions flowing and mediating throughout various ion channels of the membrane. Indeed, the action of neurotransmitters are described in this very way, since such transmitters bind to receptors on the cell s membrane (Decol et al., 2008). It is further understood, however, that at a certain duration the postsynaptic potential will produce an impulse (i.e., spike) at threshold, and it is the ongoing train of such spikes (i.e., spike emissions) that will eventually be carried through at the inter-neuronal level as units of information (Gerstner and Kistler, 2002). In this respect, it is not so uncommon to conceptualize the entire transmission behavior of neurons here as the collective dynamics of a large noisy network, with both neurons and synapses as the basic elements of the network, as pointed out in (Reutimann et al., 2003). Neuronal dynamics, therefore, has had a deep history of being analyzed through various mean-field models (Gerstner and Kistler, 2002), largely because one can reflect the behavior of a population of neurons more directly with known data in neuroscience, as would be collected from diagnostic studies in the context of electroencephalogram (EEG), magnetoence-phalogram (MEG), and functional magnetic resonance imaging (fmri) techniques (Decol et al., 2008). On the other hand, simulating the passage of neuron information under such networks has already reached the standard understanding of being encoded mostly as the neuron s spiking or firing-rate, which, in turn, is measured in terms of the frequency of the neuron s action potential (Lindner and Longtin, 2004). For the purpose of this brief communication, however, the stochastic dynamics of neuron spikes will be assumed to be governed by a Fokker-Planck equation. This is justified on the basis

4 Neuronal Spiking Model 385 that neuronal dynamics are being captured in a large embedded network of simulated neurons that can be viewed as a continuous, although random, injection of current, whose statistical properties can be approximated as an ensemble or population density model. In this sense, the stochastic dynamics of neurons are governed by a stochastic differential equation, called the Langevin equation, so that the degree of randomness, as well as the fluctuations induced by the injected current, can readily be proven to reduce into a master equation for this kind of stochastic process (Gerstner and Kistler, 2002). A Fokker-Planck equation is, therefore, the appropriate density equation to use at this level of neuronal dynamics, since it is itself the appropriate master equation, or Kolmogorov differential equation, to adopt for stochastic noise-driven systems. Here the reader can refer to Cox and Miller (1965) and Risken (1989) for further details on the historical development of Fokker-Planck equations, as deemed relevant to interacting network models and population density ensembles, as well as its use in various other statistical mean-field theories and related approaches. Method of Solution Consider an ensemble of a large number of spiking neurons that are believed to populate phase-space with a density of P (V,t), where V is the membrane potential (Reutimann et al., 2003). In other words, the probability density function, P (V,t), whichdescribes the statistics of the dynamics between successive neuron spikes in the voltage-time domain, is governed by a Fokker-Planck equation, namely, P(V, t) t = σ P(V, t) V 2 [ µ (V V ] rest) P(V, t) τ m V + 1 τ m P(V, t). (1) Here, V rest is the resting potential that acts as a rigid barrier to the membrane potential V,andτ m is the passive-membrane time constant, with the mean µ and variance σ 2 characterizing the entire stochastic process per unit time. Equation (1) is also subject to the

5 386 D. J. Daniel initial condition P (V, t) = δ (V V 0 ), (2) where δ (V V 0 ) is a Dirac delta function with V 0 representing the starting potential or initial state of the neuron at the time t = 0. Aside from the fact that there are exact solutions of Fokker- Planck equations known for various ensemble models (Risken, 1989), the computational challenge here with Fokker-Planck equation methods in neuron network models originates with the actual numerical implementation of the delta function in Equation (2). To the authors knowledge, there are no known exact solutions to Equation (1), apart from numerical solutions, represented in the neuroscience literature. For this reason, the present author has developed a method in Daniel (2008) (referred to here as I), which is capable of producing the exact solution to Equation (1), and, at the same time, satisfies the initial condition in Equation (2) the aim of which is to further the numerical implementation of Fokker-Planck equations in computational neuroscience. Following Sections 4 and 5 of I, Equation (1) is first transformed into the new variables y a 0 b 0 V, with a 0 = µτ m + V rest and b 0 = 1. (3) τ m τ m The Fokker-Planck equation, defined in Equation (1), now reads P (y, t) t = σ 2 b P (y, t) P(y, t) + b y 2 0 y + b 0 P (y, t), (4) y where, for the actual purpose of producing an exact solution to Equation (1) or Equation (4), asymptotic boundary conditions are assumed, so that lim P (y, t) = 0. (5) y Next, one recalls that the formal definition of the Dirac delta function is given by 1 δ (V V 0 ) = lim ε πε e (V V 0 ) 2 4ε, (6)

6 Neuronal Spiking Model 387 so that on account of the Gaussian expression appearing on the right-hand side of this equation being known to possess an orthogonal series expansion, it is reasonable then to seek a solution to the Fokker-Plank equation of Equation (4) in the form of the ansatz P (y, t) = n=0 a n (t) ( ) H n ipy, (7) n! where H n (ipy) is a Hermite polynomial of order n, a n (t) are time-dependent coefficients to be determined, and i is a pure imaginary complex number. One now makes a careful choice for the arbitrary constant p appearing in Equation (7), namely, p =±1/(σ b 0 ), and then substitute the ansatz in Equation (7) into the Fokker-Planck equation of Equation (4), which will be satisfied by the ansatz, provided one solves a n (t) as a n (t) = a n (0) e b 0(n+1)t. (8) The coefficients a n (0) here are determined simply by matching the ansatz in Equation (7) with the initial condition at time t = 0, given in Equation (5). As shown in Section 5 of I, the result is a n (0) = pe α 0 γ 2 0 y πεα 0 1 γ n H n (±iγ 0 py 0 )(±γ 0 ) n, (9) where α 0 = p b 2 0 ε, γ 0 = 1 2b 0 εα0, y 0 = a 0 b 0 V 0. (10) Using well-know summation formulas involving the product of two Hermite polynomials (e.g., see Magnus et al. 1966), one can substitute the result of Equation (9) into the ansatz of Equation (7) and evaluate the infinite sum to produce the exact

7 388 D. J. Daniel solution in closed form (taking the upper sign): P (y, t) = e α 0 γ 2 0 y εα 0 1 γ 2 0 e b 0t 2 n H n (±iγ 0 py 0 ) H n (ip y) n! n=0 (±γ 0 e b 0t ) n = pb γ (t) 1 γ 0 e (p 2 +α 0) γ0 2 y π 1 γ 2 [ (t) ( ) ] 2 exp y γ 0 y 0, γ (t) p 2 γ 2 (t) 1 γ 2 (t) (11) which completes the goal of obtaining the exact solution to the Fokker-Planck equation of Equation (4). Furthermore, one can normalize the solution in Equation (11) by invoking the requirement P (V, t) dv = 1, from which one obtains the normalized solution, up to a complex phase factor of e iπ/2 (which has been absorbed into the normalization constant), as P (y, t) = p π [ γ (t) γ 2 (t) 1 exp p 2 γ 2 (t) 1 γ 2 (t) ( y γ ) ] 2 0y 0, γ (t) (12) where γ (t) = γ 0 e b 0t. For completeness, it is worth justifying that the term γ 2 (t)/(1 γ 2 (t)) in the exponential argument of Equation (12) will always be negative for t > 0 provided that εis chosen suitably small enough. In fact, γ 2 (t)/(1 γ 2 (t)) 1ast. Results and Conclusion A plot of Equation (12) was carried out in the present research using a Fortran 90/95 algorithm documented by the present author in an appendix in I. The resulting plot is shown in Figure 1, in which the values for the dynamical quantities appearing in Equation (12) correspond with the same values adopted in Figure 1

8 Neuronal Spiking Model 389 FIGURE 1 Numerical plot of the solution P (y, t) of the Fokker-Planck equation specified by P (y, t) in Equation (12) of the text. Neuron parameters are: τm = 20 ms, V0 = 70 mv, Vr e s t = 70 mv, Vmin = 80 mv, µτm = 14.5 mv and σ 2 τm = 10.2 mv2. of the article by Reutimann et al. (2003). There is no question that there is a remarkable likeness between the exact solution displayed in Equation (10) and the numerical solution (Figure 1) of Reutimann et al., so that it is safe to say that the present work does, indeed, provide positive evidence toward, as well as supports, the neuroscience data results of Reutimann et al. Of course, the slight differences between the exact solution presented here and the numerical solution of Reutimann et al. is attributed to the fact that Reutimann et al. also impose additional boundary conditions in their numerical work on the Fokker-Planck equation of Equation (4), which are in contrast to those used in the present work, given by Equation (5). Finally, on account of the definition of the Dirac delta function in Equation (6), one should at least attempt to take the limit ε 0+ in Equation (12) to extract the true nature of the exact

9 390 D. J. Daniel solution, which is possible, only if the limit ε 0 + exists. Since, γ (t) e b 0t in the limit ε 0 +, one clearly ascertains the true solution as P exact (y, t) lim P (y, t) ε 0 + = p e b [ 0t p 2 π e 2b 0t 1 exp e 2b 0t 1 e 2b 0t ( y y ) 2 ] 0. (13) e b 0t Before closing, it is worth recalling again that the objective of this research is, of course, to assess the numerical accuracy incurred when implementing Fokker-Planck equations subject to a Delta function as the initial condition in computational studies of this sort, which is now made more transparent through the comparison between the ε-dependent solution obtained in Equation (12) and the exact solution of Equation (13). Acknowledgements The author wishes to express his gratitude to Professor S.-Y. Goh and his Ph.D. student, Ms. S.-C. Chan, at the Faculty of Engineering and Science, University Tunku Abdul Rahman, for invaluable discussions related to this work, as well as for communicating to the author their preliminary results. References Cox, D. R., Miller, H. D. (1965). The Theory of Stochastic Processes, Boca Raton: Chapman & Hall/CRC. Daniel, D. J. (2008). Physics Research Notes on the Langevin Equation for a Neuronal Spiking Model, University Tunku Abdul Rahman, Faculty of Science and Engineering, unpublished notes, available at com/ djdaniel/pdf files/langevineq2 V3.pdf. Decol, G., Jirsa, V. K., Robinson, P. A., Breakspear, M., Friston, K. (2008). The dynamic brain: From spiking neurons to neural masses and cortical fields. PLoS Comput. Biol. 4:1 35. Gerstner, W., Kistler, W. M. (2002). Spiking Neuron Models: Single Neurons, Populations, Plasticity. Cambridge: Cambridge University Press.

10 Neuronal Spiking Model 391 Lindner, B., Longtin, A. (2004). Effect of an exponentially decaying threshold on the firing statistics of a stochastic integrate-and-fire neuron. J. Theor. Biol. 232: Magnus, W., Oberhettinger, F., Soni, R. P. (1966). Formulas and Theorems for the Special Functions of Mathematical Physics (3rd Ed.), Berlin: Springer. Reutimann, J., Giugliano, M., Fusi, S. (2003). Event-driven simulation of spiking neurons with stochastic dynamics. Neural Comp. 15: Risken, H. (1989). The Fokker Planck Equation: Methods of Solution and Applications (2nd Ed.). Berlin: Springer-Verlag.