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PROCEEDINGS OF THE 3RD INTERNATIONAL CONFERENCE ON APPLIED MATHEMATICS, SIMULATION, MODELLING (ASM'09) PROCEEDINGS OF THE 3RD INTERNATIONAL CONFERENCE ON CIRCUITS, SYSTEMS AND SIGNALS (CSS'09) Doubly stochastic processes: an approach for understanding central nervous system activity Janet A. Best constitute part of the neural signal [8]. When the neuronal activity is stochastic, the transitions themselves may also be stochastic, for example as fluctuations in spike threshold. Doubly stochastic processes are a natural tool for understanding many types of neuronal processing in which (i) neurons fire irregularly, (ii) the firing rate changes in response to inputs, and (iii) there are underlying state changes (likely changing the stochastic firing rate of inputs) that can be modeled stochastically. Likely applications include firing of globus pallidum neurons [9] and modifying synapses during learning [10]. Doubly stochastic processes themselves are not new; they have been investigated in apurely mathematical context [11][13]. While they have been applied in computational neuroscience [14], [15] such use has been relatively infrequent. Even if one agrees that doubly stochastic processes are a natural tool in these research areas, two natural questions arise. First, what kind of evidence in data would be necessary to prove that these processes are doubly stochastic? Secondly, if they are doubly stochastic, how does that help one understand neural processing and biological function? Below we describe new work on sleep-wake transitions where this research program has been carried out and where both questions have been answered. Abstract— In this paper we argue that doubly stochastic processes are a natural tool for understanding certain types of information processing in the central nervous system. Doubly stochastic processes themselves are not new and have been investigated in a mathematical context; however, they have not been widely applied in neuroscience. We begin by pointing out that the brain is very stochastic: both individual neurons and physiological neural networks exhibit a variety of stochastic firing patterns. We then consider transitions from one firing pattern to another in individual cells and argue that the transitions themselves are likely to be stochastic. Even if one agrees that doubly stochastic processes are a natural tool in these research areas, two natural questions arise. First, what kind of evidence in data would be necessary to prove that these processes are doubly stochastic? Secondly, if they are doubly stochastic, how does that help one understand neural processing and biological function? We describe new work on sleep-wake transitions where this research program has been carried out and where both questions have been answered. Keywords— Doubly stochastic, Poisson process, power law, sleep-wake. I. INTRODUCTION T HE brain is a very stochastic place: both individual neurons and physiological neural networks exhibit a variety of stochastic firing patterns [1] and stochastic processes have been widely applied to model phenomena in neurobiology. Of course, the effects of stochasticity can also vary widely and may depend upon details of the system, resulting for instance in stochastic resonance [2], increased reliability [3] or diminished stability [4]. Though we are still learning to interpret neural codes and still debating the roles of irregularity, what is clear is that the healthy brain has ways to regulate variability in neuronal activity: the spikes of an individual neuron or of a population of neurons can be very regular or very irregular [5]-[7]. In this short paper, we focus on the irregular case: instances in which the activity of a cell or of a population of neurons is well-modeled by a stochastic process. We interpret the fact that brain has the ability to avoid such irregularity as an indication that the apparent stochasticity may have some role and thus merits attention. In many cases, transitions between distinct activity patterns II. A DOUBLY STOCHASTIC MODEL OF SLEEP-WAKE REGULATION A. Developmental Changes in Sleep-Wake Regulation The physiological complexity of sleep invites mathematical approaches and contributions: mathematical models of the neural interactions may help untangle causality or at least plausibility. A number of mathematical models have been developed to address aspects of sleep regulation. The basic neural circuitry underlying sleep-wake transitions is complex and remains to be fully identified. A reduced circuitry regulates sleep in infant mammals compared to adults: infant sleep-wake cycles arise from a brain stem network; during maturation, a gradual strengthening of modulatory inputs from more rostral areas parallels a consolidation of sleep and wake states into longer episodes [16],[17]. Blumberg and collaborators have studied these developmental changes in several model species, particularly in young rats. They observe that sleep and wake episodes are very brief for newborn animals, with exponentially-distributed Manuscript received December 15, 2009. The research was supported by the Air Force Office of Scientific Research grant FA9550-06-1-0033 and by the National Science Foundation under agreement 0112050. J. A. Best is with the Mathematics Department, Ohio State University, Columbus, OH 43210 USA (e-mail: [email protected]). ISBN: 978-960-474-147-2 155 ISSN: 1790-5117 PROCEEDINGS OF THE 3RD INTERNATIONAL CONFERENCE ON APPLIED MATHEMATICS, SIMULATION, MODELLING (ASM'09) PROCEEDINGS OF THE 3RD INTERNATIONAL CONFERENCE ON CIRCUITS, SYSTEMS AND SIGNALS (CSS'09) data as described above. Based on inferences from lesion and other experiments (see for example [16]), the model has mutually-inhibitory interactions between the wake-active dorsolateral pontine tegmentum (DLPT) and the sleep-active nucleus pontis oralis (NPO), and the firing of neurons in each nucleus are modeled as inhomogeneous Poisson processes. The power law arises due to a mechanism involving strengthening of an excitatory connection to DLPT from the locus coeruleus (LC) [22]. Fig. 1 depicts this network. duration t. In other words, letting p(r) denote the probability density function of durations between r and r+dr, so that P(t) ≡ ∫ ∞ t p(r)dr is the cumulative distribution of episode durations, they observe that P(t)~exp(-t/τ). Here, τ is the expected value of the distribution and the characteristic time scale of the behavioral state; typically τsleep ≠ τwake. The exponential shape of these two distributions persists during the first two postnatal weeks, with increasing time constants τ reflecting consolidation into longer episodes. Around the beginning of the third postnatal week, an additional process C. Doubly Stochastic Point Process Models and Bifurcation Theory We model the neuronal network governing sleep-wake as a set of three interacting cell populations. The firing times of the ith population are given by a Poisson process with intensity λi(t) which satisfies the stochastic differential equation n λi′(t) = f i ( λi ) + ∑ gij ( λi )∑δ (T jk − t) j j=1 for 1 ≤ i ≤ 3. Since λi(t) is itself stochastic, the overall process is a doubly stochastic process. Here f gives the firing rate in the absence of inputs, and gij describes the excitatory or inhibitory impact of population j on population i. The model describes the time evolution of the intensity process λi for each population. The intensity is dependent on the random variables Tjkj which are the times of firing of the j-th population. Because the full stochastic system is difficult to analyze, we have derived a system of ordinary differential equations that approximate (in a manner that can be stated precisely) the stochastic system for sufficiently short periods of time. Briefly, Fig. 1. The brain stem network. Arrows represent excitatory connections, circles denote inhibitory connections. Dashed arrows represent inputs from unspecified rostral nuclei, responsible for bout consolidation. E[ λ ′ | Λ t ] = f (λ (t)) + ∑ g (λ(t))λ (t) j j (2) j ∈input becomes evident, transforming the distribution of wake episodes such that the probability of ending a wake episode decreases across the duration of the episode. By postnatal day 21, the length of wake episodes are well-approximated by a power law distribution, P(t) ~ t-α, as has been observed in adult mammals of several species [18]. The exponential distribution for sleep bouts appears to remain throughout the life of the animal. where the expectation is conditioned on Λt, the history of the process up to time t. Tuckwell and others have studies similar questions in the case of stochastic diffusion; in their case, conditioning on the past was unnecessary [23]. If we make the assumption that λ'(t) = E[λ' | Λt], we obtain a system of three deterministic equations that has a bifurcation diagram as shown in Fig. 2. We now address the significance of the different parts of the diagram. The vertical axis of the diagram corresponds to the firing rate of a wake-active cell population during wakefulness. The bifurcation parameter, on the horizontal axis, α represents the maximal extent of positive feedback during an individual wake bout; α increases with age. For animals less than 14 days old, α<αSN, so that there is only one (stable) equilibrium for the firing rate, corresponding to a constant probability of terminating the wake bout and therefore an exponential distribution of wake bout durations. As α increases beyond αSN, the system begins to have two stable equilibria for firing rate, separated by an unstable fixed point. As a wake bout begins, the firing rate is near the lower stable fixed point; however, as wakefulness continues, positive feedback increases, resulting in a biased random walk towards the upper stable fixed point. Note that, if the wake bout continues long enough, the firing rate can remain near the B. Noisy Data; Mathematical Model All biological data contain variation from many sources, resulting in a noisy appearance. Yet the data on sleep and wake bout lengths are not merely noisy. Data showing, for example, that the distribution of wake bouts in adult rats is well fit by a power law (over three orders of magnitude) indicate that there are underlying stochastic processes at work. Therefore we have been investigating the nature of stochastic mechanisms that may be giving rise to the observed data. The underlying scientific rationale is that the mechanisms causing the bout distribution are exactly the mechanisms of transition between sleep and wakefulness that we seek to understand. We have constructed mathematical models for the developing infant neural circuitry [19]-[21]. These models, summarized below, capture well the features of experimental ISBN: 978-960-474-147-2 (1) kj 156 ISSN: 1790-5117 PROCEEDINGS OF THE 3RD INTERNATIONAL CONFERENCE ON APPLIED MATHEMATICS, SIMULATION, MODELLING (ASM'09) PROCEEDINGS OF THE 3RD INTERNATIONAL CONFERENCE ON CIRCUITS, SYSTEMS AND SIGNALS (CSS'09) analyze mathematically, we have illustrated here that pairing with deterministic systems can aide in analysis, and we continue to work on the mathematical theory to support and extend this approach. Depending on one’s purpose, it may be adequate to approximate a doubly stochastic process with a simpler inhomogenous stochastic process or a mixture of simpler stochastic processes though such an approximation may forfeit the opportunity of a mechanistic connection for the model. However, for the model described above we have been able to develop a mixed Poisson process model that yields complementary biological insights; see [21] for details. A natural question concerns what sort of evidence would support the occurrence of doubly stochastic processes in the central nervous system. We note that other authors [14] have also found, in a comparison of stochastic models, that doubly stochastic models fit the irregular activity of some neurons better than several alternative stochastic models. In this paper we have focused only upon doubly stochastic Poisson processes. However, one must expect that many other types of doubly stochastic processes occur in the central nervous system. Another natural class of processes that may be described as doubly stochastic is that of stochastic processes on random graphs; such processes may also have significant applications in neuroscience. In the event that a neuronal network changes stochastically, for example synaptic plasticity in learning, the activity on the network may be a doubly stochastic process. In related work, we are studying the probabilistic spread of neuronal activity through a random Fig. 2. Bifurcation diagram for the deterministic system. Here λ network. represents the firing frequency of DLPT during wakefulness; α represents the strength of mutual excitation between DLPT and LC and For sleep-wake cycling described above, the doubly increases with development. A saddle-node bifurcation occurs at αSN. stochastic model provides a framework for understanding how The arrow indicates the increase in intensity λ during a wake bout, due the observed data can arise from the interactions of groups of to mutual excitation. neurons. In particular, the model suggests how the postnatal distribution will depend upon details of the stochastic process development of the neuronal circuitry may drive the changes in governing the biased random walk. In ongoing work, we are sleep-wake dynamics observed through the shifting shape of studying the conditions under which this walk gives rise to bout distributions . power laws or other prescribed distributions for bout length. This work has significance in a number of areas unrelated to sleep-wake cycling. Intervals of behavior and their distribution have been long studied by ethologists [25]; typical REFERENCES approaches have involved fitting the distribution with a [1] H. C. Tuckwell, Stochastic Processes in the Neurosciences. Philadelphia, PA: Society for Industrial and Applied Mathematics, mixture of exponential processes. While a mixture of 1989. exponentials is well-known to be able to approximate other [2] A. Longtin, “Stochastic resonance in neuron models,” Journal of distributions including power law, such an approach is Statistical Physics, vol. 70, pp. 309-327, 1993. statistical only and does not shed light on the mechanisms [3] S. Tanabe, “Noise-enhanced neuronal reliability,” Phys Review E, vol. 64, 041904, 2001. generating the observed behavior. Our goal is to develop [4] E. T. Rolls, M. Loh, G. Deco, G. Winterer, “Computational models of methods for investigating the biological mechanisms. schizophrenia and dopamine modulation in the prefrontal cortex,” upper fixed point, resulting again in an additional set of exponentially-distributed wake bouts, but these with larger mean than those corresponding to the lower fixed point. In fact, these two ranges of exponentially-distributed wake bouts (separated by a range with power-law distribution) have recently been reported experimentally in adult animals [24]. The distribution of wake bout durations corresponding to lengths between these two exponential ranges will clearly have a heavy tail compared to the shorter-duration exponential bouts. The precise shape and width of this part of the [5] III. CONCLUSION In this paper we have argued that activity in the brain frequently appears stochastic and that in fact doubly stochastic mathematical models are often a natural tool that can provide real insights into biological mechanisms. We should note in this context that deterministic systems can also give rise to irregular activity [9]. Though doubly stochastic processes can be difficult to ISBN: 978-960-474-147-2 [6] [7] 157 Nature Reviews Neurosci, vol. 9, pp. 696-709, 2008. Y. Prut, S. I. Perlmutter, “Firing properties of spinal interneurons during voluntary movement. 1. state-dependent regularity of firing,” The Journal of Neurosci, vol. 23, pp. 9600-9610, 2003. S.-L. Shin, S. Rotter, A. Aertsen, E. De Schutter, “Stochastic description of complex and simple spike firing in cerebellar Purkinje cells,” European J of Neurosci, vol. 25, pp. 785-794, 2007. M. Rudolph, A. Destexhe, “The discharge variability of neocortical neurons during high-conductance states,” Neuroscience, v. 119, pp. 855-873, 2003. ISSN: 1790-5117 PROCEEDINGS OF THE 3RD INTERNATIONAL CONFERENCE ON APPLIED MATHEMATICS, SIMULATION, MODELLING (ASM'09) PROCEEDINGS OF THE 3RD INTERNATIONAL CONFERENCE ON CIRCUITS, SYSTEMS AND SIGNALS (CSS'09) [8] J. Deppisch, H.-U. Bauer, T. Schillen, P. Konig, K. Pawelzik, T. Geisel, “Alternating oscillatory and stochastic states in a network of spiking neurons,” Network: Computation in Neural Systems, vol. 4, pp. 243257, 1993. [9] J. A. Best, C. Park, D. Terman, C Wilson, “Transitions between irregular and rhythmic firing patterns in excitatory-inhibitory neuronal networks,” J Computational Neurosci, vol. 23, pp. 217-235, 2007. [10] H. S. Seung, “Learning in spiking neural networks by reinforcement of stochastic synaptic transmission,” Neuron, vol. 40, pp. 1063-1073, 2003. [11] D. R. Cox, “Some statistical methods related with series of events,” J Royal Statistical Soc. B, v. 17, pp. 129-164, 1955. [12] J. Mecke, “Eine characteristische Eigenschaft der doppelt stochastischen Poissonschen Prozesse,” Z. Wahrscheinlichkeitstheorie verw. Geb., vol. 11, pp. 74-81, 1968. [13] J. Grandell, Doubly Stochastic Poisson Processes. Berlin: SpringerVerlag, 1976. [14] S. Shinomoto, T. Yasuhiro, “Modeling spiking behavior of neurons with time-dependent Poisson processes,” Physical Review E, vol. 64, 041910, 2001. [15] B. E. A. Saleh, M. C. Teich, “Multiplication and Refractoriness in the Cat's Retinal-Ganglion-Cell Discharge at Low Light Levels,” Cybernetics, vol. 52, pp.101-107, 1985. [16] K. Karlsson, A. Gall, E. Mohns, A. Seelke, M. Blumberg, “The neural substrates of infant sleep in rats,” PLOS Biology, vol. 3, pp. 0891-0901, 2005. [17] M. Blumberg, A. Seelke, S. Lowen, K. Karlsson, “Dynamics of sleepwake cyclicity in developing rats,” PNAS, vol. 102, pp. 14860-14864, 2005. [18] C. C. Lo, T. Chou, T. Penzel, T. Scammell, R. Strecker, H. Stanley, P. Ivanov, “Common scale-invariant patterns of sleep-wake transitions across mammalian species,” PNAS, vol. 101, pp. 17545-17548, 2004. [19] J. Best, C. Diniz Behn, G. Poe, V. Booth, “Neuronal models for sleepwake regulation and synaptic reorganization in the sleeping hippocampus,” Journal of Biological Rhythms, vol. 22, pp. 220-232, 2007. [20] B. Joshi, “A doubly stochastic Poisson process model for wake-sleep cycling,” Ph.D. dissertation, Dept. Math., Ohio State University, Columbus, Ohio, USA, 2009. [21] B. Joshi, J. Best, “A doubly stochastic model for the development of sleep-wake regulation,” unpublished. [22] A. Gall, B. Joshi, J. Best, M. Blumberg, “Developmental emergence of power-law wake behavior depends upon the functional integrity of the locus coeruleus,” Sleep, vol. 32, pp. 920-926, 2009. [23] R. Rodriguez, H. Tuckwell, “Statistical properties of stochastic nonlinear dynamical models of single spiking neurons and neural networks,” Physical Review E, vol. 54, pp. 5585-5590, 1996. [24] S. Simasko, S. Mukherjee, “Novel analysis of sleep patterns in rats separates periods of vigilance cycling from long-duration wake events,” Behavioral Brain Research, vol. 196, pp. 228-236, 2009. [25] P. W. Colgan, Quantitative Ethology, New York: John Wiley & Sons, 1978. ISBN: 978-960-474-147-2 158 ISSN: 1790-5117