BioSystems 33 (1994) l-16
A stochastic model for neuronal bursting
Arnold0 Frigessi* a, Petr L6nskib, Angela B. Mariottoc*d
aLaboratorio di Statistica, Universitri di Vene:ia. Ca Foscari, Dorsoduro 3246, I-30123 Vene:ia, Italy
blnstitute o_f Physiology. Academy oj Sciences, Videriski 1083. 142 20 Praha 4-KrC: C:ech Republic
cistituto per le Applica:ioni de1 Calcolo Mauro Picone. CNR. Viale drl Policlinico 137, l-00161 Rome. Ital) lstituto Superiore di Sanita: Laboratorio di Epidemiologia e Biostatistica, Viale Fegina Elena 299. I-00161 Ram;. Ita!v
Received 14 July 1993
A new stochastic model for bursting of neuronal tiring is proposed. It is based on stochastic diffusion and related to the first passage time problem. However, the model is not of renewal type. Its form and parameters are physiologi- cally interpretable. Parametric and non-parametric inferential issues are discussed.
Keywords: Neuronal model; Membrane potential; Bursting: Stochastic process; Statistical inference
Stochastic diffusion neuronal models represent one of the most advanced and successful descrip- tions of the membrane electric potential of a neu- ron. Most of these models presented in the literature have a common property: they generate spike trains that are of renewal type and whose interspike intervals (ISIS) have a unimodal proba- bility density function. In experiments with spon- taneous or driven neuronal activity, cases are often encountered for which one or both of these condi- tions are not met. The main aim of this paper is to propose a very simple model, which is based on stochastic diffusion and is physiologically inter- pretable, that allows these drawbacks to be over-
* Corresponding author.
come. It describes the so called bursting activity of a neuron. The model is intended mainly for a neu- ron within a neural network; however, the wiring of the net is not specified.
Bursting is a very frequent type of neuronal behaviour. A broad class of quite different phe- nomena are classified as bursts. The common fea- ture consists in a sequence of short ISIS separated by one (or a few) long interval(s). The next sec- tion of this paper tries to put some order in an ample and scattered literature on this topic. We survey and present some of the main models and their applications. Physiological objections are pointed out on the basis of selected experimental findings. General features of bursting are col- lected, in order to be incorporated in our new model. In Section 3 we define the model, in terms of stochastic diffusions and their first hitting times
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to certain thresholds. Illustrative examples of the model simulation are presented. Although our model is certainly a very rough approximation of the true membrane dynamics, its form and all its parameters possess a physiological interpretation. For this reason not only the qualitative character- istics of our model are compared with experimen- tal data, but also the inferential problems are considered. The mathematical model allows us to construct statistical parameter estimation pro- cedures as described in Section 4.
2. Bursting neuronal activity
Bursting pacemaker neurons have been often described in invertebrate animals. In these it has been proven that the burst arises from an endogen- ous pacemaker mechanism rather than from exter- nal synaptic input. This type of activity can be found, for example, in abdominal ganglia of slugs and snails, and in cardiac pacemaker cells or stomatogastic ganglion cells of crustacea. Katayama (1973) presented several other examples of this bursting activity with quantitative descriptions of the experimental results. Although our aim is to analyse a non-endogenous activity, which is not perfectly stereotyped as in the cases mentioned above, we will mention it here. The reason is, on the one hand, to stress the distinction from the stochastic bursting, and on the other hand, to point out features also applicable in our problem. There is greater experimental evidence about the bursts and their discharge patterns. Also, a very precise formulation of the terms which are in use for the mathematical models exists (Carpenter, 1981). Nevertheless, we mention this type of bursting not only because of the formal reason, but also because of the the possibility that the burst patterns in these neurons could be also under the control of some exogenous factors (see, for ex- ample, Pin and Gala, 1983).
Some common features in bursting patterns can be found. Often the length of the interburst inter- val depends on the number of spikes in the pre- ceeding burst; the first IS1 within the burst tends to correlate with the interburst interval preceeding the burst. Clearly, the instant of spike generation need not be of renewal type. Also, the number of
spikes and the ISIS within the burst are variable. There are several spike pattern classifications. For the parabolic bursters the spiking frequency first increases and then decreases. For some others, the falling or the increasing phase is not necessarily present. The mathematical modelling of endogen- ous pacemaker bursting presented in the literature is based on the analysis of Hodgkin-Huxley type deterministic models (e.g., Plant, 1981). Here one may introduce an aperiodic discharge by means of the :heory of chaos (Chay and Rinzel, 1985) or by introducing noise into the *Hodgkin-Huxley sys- tem (Carpenter, 1981). Chay (1990) studied a deterministic model in which a voltage-activated Ca2+ channel inactivates slowly upon hyper- polarization. In her model, bursting is controlled by the relaxation time constant of the K+ channel that is activated by voltage. A sigmoidal shape (Boltzmann form) is assumed for the relaxation curve with respect to the voltage.
The spike trains recorded in the central nervous system of mammals sometimes have bimodal or multimodal IS1 histograms and exhibit bursting behaviour, for example the pyramidal cells of the hippocampus treated by penicillin, (Traub and Llinas, 1979), and the neurons of the mesencepha- lit reticular formation (Yamamoto and Nakahama, 1983; Yamamoto et al., 1986; Lansky and Radii, 1987). Bishop et al. ( 1964) presented experimental data of bursting character recorded in lateral ge- niculate neurons of anaesthetized cats. Legendy and Salcman (1985) studied in detail bursting phe- nomenon in neurons of the striate cortex of anaesthetized cats. They called burst an epoch of elevated discharge rate. For this purpose they defined, approximately, a burst as a minimum of three spikes for which the largest spacing was less than a half mean ISI (for precise definition see the cited paper). The size of burst is thus at least three for these authors. Abeles et al. (1990) investigated high-frequency bursts of activity recorded from the cortical areas. These authors also applied the term burst in a highly restrictive sense and they concluded that the probability of observing a burst in one neuron was not affected by the fact that another adjacent neuron emmited a burst. Many other experimental data of this type have been col- lected.
A. Frigessi et al. / BioSystems 33 (1994) I - 16 3
Theoretical models for the bursting phenomena often based on the assumption that the neuron alternates between two states. The OFF state. dur- ing which no spike can be generated, and the ON state, when spikes are produced. Smith and Smith (1965) presented the classical model of this type producing a mixture of two exponentials as the ISI distribution. Thomas (1966) introduced a mode] for a bursting neuron based on the intraneuronal mechanism proposed by Burns (1955). The model is similar to a branching Poisson process and analogously to the model of Smith and Smith (1965) it can be formulated as a two-state semi- Markov process. Bursting in mesencephalic reticular formation neurons was recently studied by Griineis et al. (1989). A branching Poisson pro- cess (also called the Bartlett-Lewis process) is used in their description; the difference between their model and that of Thomas (1966) is that Thomass model does not permit overlapping of subsidiary processes. There, in the branching Poisson process, a series of primary events is assumed to form a stationary Poisson process. The secondary process consists of a random number of spikes and intervals that follow a gamma distribu- tion. These assumptions permit the authors to compare the model with experimental data. For our model two of their conclusions are important. The first one is a Markov-dependency in the ISI structure, suggesting that the neuron output pro- cess holds memory about the input for a time which corresponds to the duration of burst. The second conclusion concerns the distribution of the intervals between spikes within a burst, which turns to be of gamma type, with a shape close to the inverse Gaussian distribution.
Ekholm (1972) modelled this type of neuronal activity as a generalized semi-Markov process, named pseudo-Markov process. In an extensive paper by de Kwaadsteniet (1982), the model pro- posed by Ekholm is called a semi-alternating renewal model and is analysed in more detail. Despite a close resemblance of the output of the model with the experimentally measured data, a serious objection must be raised: in the model for- mulation the number of spikes in the burst is deter- mined a priori when the burst starts. This is physiologically hardly possible, because the num-
her of spikes in the burst is determined by the dynamics of the neuronal input during the whole bursting period.
The selective interaction models and the models with subthreshold interaction (for a survey see Holden 1976; Lansky, 1983a) are also aimed at describing bursting activity. The first of them seems to ignore relevant physiological properties by not taking into account any subthreshold inter- actions between excitation and inhibition, or more generally, any form of temporal facilitation. This defect is removed in the models with subthreshold interaction; however, there are also some objec- tions in this case. Namely, no spontaneous decay is included ,in the membrane behaviour and the discretization of the membrane potential is too rough a simplification.
Kohn (1989) introduced a model neuron divided into two compartments: (I) the dendritic tree and the cell body, and (2) the trigger zone. Simulating this model he observed bursting activity. A se- quence of action potentials was defined as a burst if at least six consecutive action potentials occured with corresponding interspike intervals less than the mean interval divided by 2.5. The bursting was always reflected by positively correlated interspike intervals and their variance was high with respect to the mean. Analogous results were predicted in Rospars .and Lansky .( 1993) for a model neuron with partial reset of the membrane potential. These two model neurons are endogenous bursters since the bursting does not arise due to the modu- lation of the input but due to the internal proper- ties of the neurons.
Holden (1976) surveyed an alternative approach to bursting discharge patterns. It is based on a numerical and simulated investigation of two leaky neuronal integrators with mutual inhibitory coupling. The output of the model shows the multimodal histogram of ISIS, though the inputs are of temporally homogeneous Poisson type. Pairs of neurons,. reciprocally coupled by in- hibitory synapses, also give rise to typical patterns of alternating bursts A see the study by Perkel and Malloney (1974). Zeevi and Bruckstein (198 I) and Bruckstein and Zeevi (1985) analysed a genera] neural encoding model based on an integrate-and- fire at threshold scheme. The main feature of their
4 A. Frigessi et af. / BioSys~ems 33 (1994) I-16
model is that it takes into account that the mem- brane integrates an effective ionic current, which depends not only on the input generator current but also on the output-dependent self-inhibition feedback. This type of feedback can be thought of as a source of systematic variability, which can be incorporated into the class of stochastic bursting model that we describe next.
Recently the rythmical synchronization of the neurons in the visual cortex was experimentally studied (Eckhom et al., 1988; Gray and Singer, 1989). It is hypothesized that lower-order neurons can produce an oscillation of activity in a phase as a response to specific inputs. The resulting post- synaptic potentials in the target cells produce a large oscillation in their membrane potential. At the peak of this oscillation, the membrane poten- tial of the higher order neuron would fire a high- frequency burst of spikes (Styker, 1989). Theo- retically, it corresponds to the periodical input to a higher-order model neuron. It is well known that such periodical input can, for appropriately chosen parameters, produce high-frequency firing during its elevated level while it remains almost si- lent during the lower phase of the input (Gummer, 1991a,b; L6nsky et al., 1992). This is one of the alternative mechanisms to the inhibitory feedback which can be a source of bursting in a neural net- work. Both these cases are examples of input con- trolled bursting.
3. The model
3.1. General description The neuronal output may be considered as a
realization of a point process. This process is generated in most of the neuronal models by the first passages of a stochastic process, called mem- brane potential or generator potential, through a threshold potential, usually (but not always) deter- ministic. During bursting the underlying generator potential changes according to the state of the ac- tivity. The neuronal behaviour is epitomized by a time-continuous one-dimensional real valued stochastic process describing the membrane poten- tial. Additionally, there are again two alternating states in which the neuron can be. These states are called bursting and inactive periods. We assume
that the parameters in the model can change only at the moment of spike generation but not during the ISI. This &mplification is reasonable and allows us to avoid unnecessary mathematical com- plications. Hence, the values of the parameters, which are reset at the beginning of the ISI, are to be considered as average values valid over the whole interval. The state of the neuron depends on the trajectory of the membrane potential from the moment of the !ast spike generation. An analogous mechanism was used by Frigessi and den Hollander (1989) for modeling a stimulated neuron.
Three constants are important for the neuron characterization. The first of them is a threshold potential A. Whenever the membrane potential reaches the value of the threshold potential A then an action potential (spike) is generated. Instan- tanously the value of the...