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Stochastic models for spike trains of single neurons
BOOK REVIEW
G. Sampath and S. K. Srinivasan, Stochastic Models for Spike Trains of Single Neurons, Lecture Notes in Biomathematics, Vol. 16, Springer, New York, 1977 188 pp., illus.; $8.30.
The heart of this book is a coherent and systematic account of that class of neuronal models that seeks to predict the statistical properties of the ongoing train of impulses produced by a neuron as a result of postulated stochastic mechanisms. The published models of this sort are described and discussed under six headings; a major contribution of the authors has been to formulate a classification of these models, bringing order into a mass of scattered and often confusing literature. The six categories are described as follows: A large class of models accounts for the statistical properties of the output impulse tram in terms of the corresponding properties of one or more input trains. The simplest such model is that of superposition: the input events are simply pooled to produce an output train, as if the neuron were a perfect, one-to-one follower of all its input channels. This model has some non-trivial consequences; in particular, if the input trains are renewal processes (independent durations of intervals between successive events), the output train is no longer a renewal process, except in the particular case of Poisson processes. The second category, that of deletion models, allows one of the input trains to be inhibitory. Each inhibitory event deletes the effect of arriving excitatory impulses for a fixed time or for a fixed number of subsequent excitatory events. In the third class of models, the pooled stream of impulses impinging on a neutron is regarded as Poisson process; each impulse is assumed to cause a small depolarization or hyperpolarization of the membrane, so that the sequence of displacements of membrane potential can be regarded as a random walk, for which a continuous diffurion model is a useful approximation. The production of an impulse is identified with crossing a boundary, so that the distribution of time intervals between firings is the solution to a first-passage-time problem. Such diffusion models of the neuron draw on the large body of diffusion theory in physics and thermodynamics; the first such model published (only later versions are cited in the volume under review) is that of Gerstein [l]. A similar kind of diffusion model, not reviewed by the authors, is one in which the membrane potential is represented by white noise, and the time-varying excitability (difference between deterministic component of membrane potential and a possibly time-varying threshold) corresponds to a moving boundary (Geisler and Goldberg [2], Enright [3]). Geiger counters and similar devices for counting randomly arriving events (particles emitted by decaying atomic nuclei) are subject to a dead time after each registration of an event; this resembles in many respects the refractory period of a neuron. The statistical theory of particle counting was expounded by Feller [4] in 1948, in a hard-to-locate paper. The body of mathematical results on particle counting based on Feller’s formulation has been used to extend the class of deletion models; such counter models comprise the fourth category of Sampath and Srinivasan. MATHEMATICAL
BIOSCIENCES
6 Elsevier North-Holland,
Inc., 1978
40, 325-32-l (1978)
325 0025~5564/78/@M@303$01.75
326
BOOK REVIEW
By combining some of the assumptions of counter models and diffusion models, another class of stochastic neuron models is obtained, in which the membrane potential is assumed to be shifted by a unit amount, either towards firing threshold if the input event is excitatory, or away from threshold if the input is inhibitory. In such discrete-state random-walk models, the time from the initial observation to firing is the first-passage time to a “boundary” representing threshold. Just as in diffusion models, if the membrane is reset after each firing to a given level of polarization, then the distribution of first-passage times approximates the distribution of intervals between impulses in the neuron. More realistic modifications of random-walk models assume that the membrane potential drifts toward resting level between synaptic events, or that the displacements caused by these events vary in amplitude. Such continuous-state random-walk models are difficult to handle mathematically, and only partial success has been achieved in obtaining explicit closed-form solutions for the interval distribution. Additional refinements to the assumptions at present require recourse to purely numerical studies, as through computer simulation. In six chapters, each devoted to one of these classes of model, the authors develop the assumptions and consequences in a systematic fashion as a sequence of theorems, illustrated with condensed but clear descriptions of particular published models. In many cases, the treatment in the book is better organized and easier to follow than that of the original paper. Overall, the consistent treatment of a wide and representative selection of papers makes the entire area of stochastic models of neuronal firing readily accessible to the student or investigator. The discussions at the end of each chapter are helpful, but they would have been far more useful if they had been expanded. An introductory set of chapters attempts to provide the reader with a basic understanding of cellular neurophysiology sufficient to follow the implications of the model. Although the intention is laudable, the task simply cannot be done within the space limitations imposed. In particular, a reader without the appropriate background would be unable to evaluate the physiological reasonableness of the assumptions made in the various models, even though the mathematical aspects are lucidly delineated; such readers have recourse to the classic introductions by Stevens [5] or Katz [6], or the more recent and wide-ranging texts by Kuffler and Nicholls [7], Kandel [8], or Bullock [9]. The final chapter, which addresses the question of realism of models, is once again useful as far as it goes, but would have benefitted from a more extensive and penetrating discussion. The coverage of the field is comprehensive, although not complete. One of the most serious omissions is that of the paper by Johannesma [lo] on diffusion models. A minor inconvenience is the division of the bibliography into two sections: “References” and “Additional references”. As in the other volumes of this series, the book is produced by photocopying typescript; this volume is gratifyingly free of misprints, and the equations are more than adequately legible. The relative importance of this kind of stochastic model with respect to other approaches to neural modeling might well be debated, especially in view of the
BOOK
327
REVIEW
sometimes drastically simplified assumptions made about neurophysiological mechanisms. Moreover, the sensitivity of predicted interval distributions to choice of model or to parameters of the model is notoriously weak, so that empirical measurements of interval histograms alone can seldom serve as a basis for discriminating among alternative hypotheses at the neuronal and subneuronal levels. However, whatever the long-term importance of this class of models may be for the experimental neuroscientist, the models themselves are intrinsically interesting as applications of stochastic processes. This book is recommended as both an introductory survey and a thoughtful, detailed, and unified account of this group of neuronal models. DONALD H. PERKEL Department of Biological Sciences Stanford Uniuersiv Stanford California REFERENCES 1 G. L. Gerstein, Mathematical models for the all-or-none activity of some neurons, IRE Trans. Info. Theory IT-8: 137-143 (1962). 2 C. D. Geisler and J. M. Goldberg, A stochastic model of the repetitive activity of neurons, Biophys. J. 6:53-69 (1966). 3 J. T. Enrignt, The spontaneous neuron subject to tonic stimulation, J. Z’heorer. BioZ. 16:54-77 (1967). 4 W. Feller, On probability problems in the theory of counters, in Studies and Essuys
Presented to R. Courant (Courant
Anniversary Volume; K. 0. Friedrichs, 0. Neugebauer, and J. J. Stoker, Eds.), Interscience, New York, 1948, pp. 105-I 15. 5 C. F. Stevens, Neurophysiologv: a Primer, Wiley, New York, 1966. 6 B. Katz, Nertx, Muscle, and Synapse, McGraw-Hill, New York, 1966. 7 S. W. Kuffler and J. G. Nicholls, From Neuron to Bruin, Sinauer, Sunderland, 1976.
E.
Mass.,
8 E. R. Kandel, Cellular Basis of Behavior, Freeman, San Francisco, 1976. 9 T. H. Bullock, Introduction to Nervous Systems, Freeman, San Francisco, 1977. 10 P. I. M. Johannesma, Diffusion models for the stochastic activity of neurons, in Neural Nefworks (E. R. Caianiello, Ed.), Springer, New York, 1968, pp. 116-144.
G. Sampath and S. K. Srinivasan, Stochastic Models for Spike Trains of Single Neurons, Lecture Notes in Biomathematics, Vol. 16, Springer, New York, 1977 188 pp., illus.; $8.30.
The heart of this book is a coherent and systematic account of that class of neuronal models that seeks to predict the statistical properties of the ongoing train of impulses produced by a neuron as a result of postulated stochastic mechanisms. The published models of this sort are described and discussed under six headings; a major contribution of the authors has been to formulate a classification of these models, bringing order into a mass of scattered and often confusing literature. The six categories are described as follows: A large class of models accounts for the statistical properties of the output impulse tram in terms of the corresponding properties of one or more input trains. The simplest such model is that of superposition: the input events are simply pooled to produce an output train, as if the neuron were a perfect, one-to-one follower of all its input channels. This model has some non-trivial consequences; in particular, if the input trains are renewal processes (independent durations of intervals between successive events), the output train is no longer a renewal process, except in the particular case of Poisson processes. The second category, that of deletion models, allows one of the input trains to be inhibitory. Each inhibitory event deletes the effect of arriving excitatory impulses for a fixed time or for a fixed number of subsequent excitatory events. In the third class of models, the pooled stream of impulses impinging on a neutron is regarded as Poisson process; each impulse is assumed to cause a small depolarization or hyperpolarization of the membrane, so that the sequence of displacements of membrane potential can be regarded as a random walk, for which a continuous diffurion model is a useful approximation. The production of an impulse is identified with crossing a boundary, so that the distribution of time intervals between firings is the solution to a first-passage-time problem. Such diffusion models of the neuron draw on the large body of diffusion theory in physics and thermodynamics; the first such model published (only later versions are cited in the volume under review) is that of Gerstein [l]. A similar kind of diffusion model, not reviewed by the authors, is one in which the membrane potential is represented by white noise, and the time-varying excitability (difference between deterministic component of membrane potential and a possibly time-varying threshold) corresponds to a moving boundary (Geisler and Goldberg [2], Enright [3]). Geiger counters and similar devices for counting randomly arriving events (particles emitted by decaying atomic nuclei) are subject to a dead time after each registration of an event; this resembles in many respects the refractory period of a neuron. The statistical theory of particle counting was expounded by Feller [4] in 1948, in a hard-to-locate paper. The body of mathematical results on particle counting based on Feller’s formulation has been used to extend the class of deletion models; such counter models comprise the fourth category of Sampath and Srinivasan. MATHEMATICAL
BIOSCIENCES
6 Elsevier North-Holland,
Inc., 1978
40, 325-32-l (1978)
325 0025~5564/78/@M@303$01.75
326
BOOK REVIEW
By combining some of the assumptions of counter models and diffusion models, another class of stochastic neuron models is obtained, in which the membrane potential is assumed to be shifted by a unit amount, either towards firing threshold if the input event is excitatory, or away from threshold if the input is inhibitory. In such discrete-state random-walk models, the time from the initial observation to firing is the first-passage time to a “boundary” representing threshold. Just as in diffusion models, if the membrane is reset after each firing to a given level of polarization, then the distribution of first-passage times approximates the distribution of intervals between impulses in the neuron. More realistic modifications of random-walk models assume that the membrane potential drifts toward resting level between synaptic events, or that the displacements caused by these events vary in amplitude. Such continuous-state random-walk models are difficult to handle mathematically, and only partial success has been achieved in obtaining explicit closed-form solutions for the interval distribution. Additional refinements to the assumptions at present require recourse to purely numerical studies, as through computer simulation. In six chapters, each devoted to one of these classes of model, the authors develop the assumptions and consequences in a systematic fashion as a sequence of theorems, illustrated with condensed but clear descriptions of particular published models. In many cases, the treatment in the book is better organized and easier to follow than that of the original paper. Overall, the consistent treatment of a wide and representative selection of papers makes the entire area of stochastic models of neuronal firing readily accessible to the student or investigator. The discussions at the end of each chapter are helpful, but they would have been far more useful if they had been expanded. An introductory set of chapters attempts to provide the reader with a basic understanding of cellular neurophysiology sufficient to follow the implications of the model. Although the intention is laudable, the task simply cannot be done within the space limitations imposed. In particular, a reader without the appropriate background would be unable to evaluate the physiological reasonableness of the assumptions made in the various models, even though the mathematical aspects are lucidly delineated; such readers have recourse to the classic introductions by Stevens [5] or Katz [6], or the more recent and wide-ranging texts by Kuffler and Nicholls [7], Kandel [8], or Bullock [9]. The final chapter, which addresses the question of realism of models, is once again useful as far as it goes, but would have benefitted from a more extensive and penetrating discussion. The coverage of the field is comprehensive, although not complete. One of the most serious omissions is that of the paper by Johannesma [lo] on diffusion models. A minor inconvenience is the division of the bibliography into two sections: “References” and “Additional references”. As in the other volumes of this series, the book is produced by photocopying typescript; this volume is gratifyingly free of misprints, and the equations are more than adequately legible. The relative importance of this kind of stochastic model with respect to other approaches to neural modeling might well be debated, especially in view of the
BOOK
327
REVIEW
sometimes drastically simplified assumptions made about neurophysiological mechanisms. Moreover, the sensitivity of predicted interval distributions to choice of model or to parameters of the model is notoriously weak, so that empirical measurements of interval histograms alone can seldom serve as a basis for discriminating among alternative hypotheses at the neuronal and subneuronal levels. However, whatever the long-term importance of this class of models may be for the experimental neuroscientist, the models themselves are intrinsically interesting as applications of stochastic processes. This book is recommended as both an introductory survey and a thoughtful, detailed, and unified account of this group of neuronal models. DONALD H. PERKEL Department of Biological Sciences Stanford Uniuersiv Stanford California REFERENCES 1 G. L. Gerstein, Mathematical models for the all-or-none activity of some neurons, IRE Trans. Info. Theory IT-8: 137-143 (1962). 2 C. D. Geisler and J. M. Goldberg, A stochastic model of the repetitive activity of neurons, Biophys. J. 6:53-69 (1966). 3 J. T. Enrignt, The spontaneous neuron subject to tonic stimulation, J. Z’heorer. BioZ. 16:54-77 (1967). 4 W. Feller, On probability problems in the theory of counters, in Studies and Essuys
Presented to R. Courant (Courant
Anniversary Volume; K. 0. Friedrichs, 0. Neugebauer, and J. J. Stoker, Eds.), Interscience, New York, 1948, pp. 105-I 15. 5 C. F. Stevens, Neurophysiologv: a Primer, Wiley, New York, 1966. 6 B. Katz, Nertx, Muscle, and Synapse, McGraw-Hill, New York, 1966. 7 S. W. Kuffler and J. G. Nicholls, From Neuron to Bruin, Sinauer, Sunderland, 1976.
E.
Mass.,
8 E. R. Kandel, Cellular Basis of Behavior, Freeman, San Francisco, 1976. 9 T. H. Bullock, Introduction to Nervous Systems, Freeman, San Francisco, 1977. 10 P. I. M. Johannesma, Diffusion models for the stochastic activity of neurons, in Neural Nefworks (E. R. Caianiello, Ed.), Springer, New York, 1968, pp. 116-144.