On some computational results for single neurons’ activity
modeling
A. Di Crescenzo
a, E. Di Nardo
a, A.G. Nobile
b, E. Pirozzi
c,
L.M. Ricciardi
d,*
aDipartimento di Matematica,Uni6ersita` della Basilicata,Potenza,Italy bDipartimento di Matematicae Informatica,Uni6ersita` di Salerno,Baronissi,SA,Italy
cDipartimento di Informatica,Matematica,Elettronica e Trasporti,Uni6ersita` di Reggio Calabria,Reggio Calabria, Italy dDipartimento di Matematica e Applicazioni,Uni6ersita` di Napoli‘Federico II’,Via Cintia,80126Naples,Italy
Abstract
The classical Ornstein – Uhlenbeck diffusion neuronal model is generalized by inclusion of a time-dependent input whose strength exponentially decreases in time. The behavior of the membrane potential is consequently seen to be modeled by a process whose mean and covariance classify it as Gaussian – Markov. The effect of the input on the neuron’s firing characteristics is investigated by comparing the firing probability densities and distributions for such a process with the corresponding ones of the Ornstein – Uhlenbeck model. All numerical results are obtained by implementation of a recently developed computational method. © 2000 Elsevier Science Ireland Ltd. All rights reserved.
Keywords:Firing distribution; Diffusion; Gauss – Markov model
www.elsevier.com/locate/biosystems
1. Introduction
Within the framework of the study of single neuron’s activity a privileged role has historically been played by mathematical models based on continuous Markov processes, better known as ‘diffusion processes’. The theoretical description of the firing process has thus been viewed as a first passage time problem (FPT) through a time-dependent threshold function, by generally assum-ing that the membrane potential after each
attainment of the threshold value is reset to a unique well-specified value. Here, we limit ourselves to pointing out that over 35 years have elapsed since the beginning of the history of neu-ronal models based on diffusion processes (cf. Gerstein and Mandelbrot, 1964). Indeed, it was then shown that for some intracellular recording the interspike intervals histograms could be fitted to an excellent degree of approximation by means of the first passage time probability density func-tion (pdf) of a Wiener process. Ever since, alterna-tive stochastic diffusion models have been proposed in the literature, aiming at refinements and embodiments of other neurophysiological
fea-* Corresponding author. Fax: +39-081-675665. E-mail address:luigi.ricciardi@unina.it (L.M. Ricciardi).
tures. The literature on this subject is too vast to be exhaustively recalled here. A comprehensive review including an outline of the appropriate mathematical techniques can be found in Riccia-rdi (1995) and in RicciaRiccia-rdi et al. (1999) and references therein.
Among the proposed neuronal diffusion models the most famous is undoubtedly the so called Ornstein – Uhlenbeck (OU) model. This includes the presence of the exponential decay of the neu-ron’s membrane potential that occurs between successive postsynaptic potentials — missing in Gernstein – Mandelbrot model — at the price, however, of a great increase of analytical com-plexity. As a consequence, related FPT problems within such a model must be approached by a variety of subtle techniques and, in general, the firing pdf is only obtained as the numerical solu-tion of a second kind Volterra integral equasolu-tion by implementation of a fast-converging al-gorithms developed by some of us (Buonocore et al., 1987).
As is well-known, diffusion neuronal models rest on the strong Markov assumption, which allows one to use various analytic methods for the FPT pdf evaluation that models the neuronal firing pdf. However, it is conceivable that, partic-ularly if a neuron is subject to strongly correlated inputs, the Markov assumption may turn out to be inappropriate, so that models based on non-Markov stochastic processes ought to be consid-ered. Some attempts along such directions may be found in Di Nardo et al. (1998a). In the present paper, we assume that the processes underlying the neuron’s model belong to the rather general Gauss – Markov class. For such processes, a new computationally simple, speedy and accurate method has been developed to obtain FPT pdf’s through time-dependent boundaries (Di Nardo et al., 1998b). Here, we report some results obtained by use of such a method to evaluate FPT pdf’s and cumulative distributions for a Gauss – Markov neuronal model — that will be further investigated in the future — representing a time-inhomogeneous generalization of OU model, in-cluding a time-dependent external input. Such results are compared with the corresponding ones for the OU model to pinpoint qualitative and quantitative diversities.
2. Neuronal models
Consistently with a frequent approach in theo-retical neurobiology literature, here we assume that the neuron’s membrane potential is described by a continuous scalar stochastic process X(t) that represents changes in the membrane potential between two consecutive neuronal firings. The threshold potential, denoted by S=S(t), with
S(t0)\x0, is assumed to be a deterministic func-tion of time. Our interest focuses on the proper-ties of the FPT random variable
T=inf{t]t0:X(t)\S(t)},
with X(t0)=x0BS(t0). Indeed, T is suitable to describe neuronal interspike intervals. The recip-rocal relationship between the firing frequency and the interspike interval naturally leads to the problem of determining the pdf of T, namely the firing probability density
g[S(t),tx0,t0]=#
#tP{T5t}.
When this function cannot be obtained analyti-cally (which is almost the rule), the analysis re-sorts to methods based on computational approximations, or on asymptotic estimates of the FPT pdf, or on the simulation of the sample paths of X(t).
2.1. Diffusion approach
A customary assumption is that processX(t) is a regular diffusion process; i.e. a Markovian pro-cess generated by the stochastic differential equation
dX(t)=A1[X(t),t]dt+A2[X(t),t]dB(t), (1) where B(t) is the standard Brownian motion and
A1(x, t) andA2(x,t) are real-valued functions of their arguments satisfying certain regularity con-ditions (cf., for instance, Karlin and Taylor, 1981). It can then be seen that the pdf of X(t) conditional on X(t0)=x0, given by
f(x,tx0,t0)= #
#xP{X(t)5xX(t0)=x0}
#f ‘infinitesimal variance’ of the process defined as
Ak(x,t)=lim
for k=1, 2. To determine f via the diffusion equation the following initial condition must be considered:
lim
t¡t0
f(x,tx0,t0)=d(x−x0).
However, as the delta initial condition is not always sufficient to determine uniquely the transi-tion pdf, suitable boundary conditransi-tions may have to be imposed (cf., for instance, Karlin and Tay-lor, 1981).
The neuronal models based on the applications of diffusion processes are predominantly time-ho-mogeneous and thus the infinitesimal momentsA1
and A2 do not depend on t. Among the time-ho-mogeneous diffusion models, ranks the above re-called OU model, characterized by infinitesimal moments
A1(x)= −1
q(x−r), A2=s
2. (3)
A diffusion approximation leading to this model and starting from a neuronal model char-acterized by an arbitrary number of excitatory and inhibitory inputs is given in Ricciardi (1976). In the absence of randomness (i.e. whens=0), for the OU model one has
dx
dt= −
1
q(x−r), x(t0)=x0. (4)
This equation expresses the spontaneous expo-nential decay of the membrane potential towards the resting potentialr. Hence,qcan be viewed as the time-constant of the neuron’s membrane (ap-proximately 5 ms).
The transition pdf for the OU model with infinitesimal moments (3), is obtained by solving the Fokker – Planck equation with the delta initial condition. One thus finds that at each timet\t0
the transition pdf f(x, tx0, t0) is normal with identifies with the solution of deterministic Eq. (4).
Unfortunately, even though the transition pdf is easily obtained, there is a lack of closed form solutions to the firing density for the OU neuronal model, as well as for other diffusion neuronal models. Hence, aiming to obtain accurate numeri-cal evaluations for the FPT pdf g in the general case of time-varying thresholds and of arbitrary time-homogeneous one-dimensional diffusion models (viz. not necessarily of the OU type), an efficient procedure has been developed in Buono-core et al. (1987), Giorno et al. (1989). This is based on the numerical solution of the following second-kind Volterra integral equation:
whereS(t) is a sufficiently smooth threshold func-tion and
Figs. 1 and 2, obtained by means of such numerical procedure, show the firing pdf’s for OU model when t0=0, x0=r= −70 and s
2=2,
2.2. Gaussian approach
By analogy with OU model, we shall now refer to correlated Gaussian neuronal models, the Gaussian nature being conceivably the result of the superposition of a very large number of synaptic inputs, as indicated in Ricciardi (1976). Along such direction, in Kostyukov (1978), Kostyukov et al. (1981) the problem of single neuron modeling was approached by restricting the analysis to Gaussian stationary processes. In the present paper, we shall refer to Gaussian processes as well. However, we shall not require stationarity, but will instead assume that their nature is Markov.
To approach our theme, let us recall some necessary notions on Gauss – Markov processes. Let {X(t), tI}, whereIis a continuous parame-ter set, be a real continuous Gauss – Markov pro-cess with the following properties:
1. m(t) E[X(t)] is continuous in I; 2. the covariance
c(s,t)E{[X(s)−m(s)][X(t)−m(t)]} is continuous in I×I;
3. X(t) is non-singular except possibly at the end points of I, where it could be equal to m(t) with probability one.
We recall that a Gaussian process is Markov if and only if its covariance satisfies
c(s,u)=c(s,t)c(t,u)
c(t,t) (6)
for all s5t5u, withs, t, u belonging to I. Well-behaved solutions of Eq. (6) are of the form
c(s,t)=h1(s)h2(t), s5t, (7) where
r(t)h1(t)
h2(t) (8)
is a monotonically increasing function by virtue of the Cauchy – Schwarz inequality, and h1(t)h2(t)\
0 because of the assumed non-singularity of the process on I. Any Gaussian process with covari-ance as in Eq. (7) can be represented in terms of the standard Wiener process {W(t), t]0} as
X(t)=m(t)+h2(t)W[r(t)], (9) and is therefore Markov.
The conditional pdf f(x, tx0, t0) of a Gauss – Markov process is a normal density characterized, respectively, by conditional mean and variance
M(tt0)=m(t)+h2(t)
h2(t0)[x0−m(t0)]
V(tt0)=h2(t)
h1(t)−h2(t)h2(t0)h1(t0)
n
, (10) witht,t0I,t0Bt. It satisfies equation of type Eq. (2) with A1(x, t) and A2(x, t) respectively, given byA1(x,t)=m%(t)+[x−m(t)] h2%(t)
h2(t)
A2(x,t)=h2
2(t)r%(t). (11)
Nevertheless, in general corresponding FPT densities are not analytically obtainable. How-ever, in analogy with the result originally
pro-Fig. 1. Plot of the firing pdf for the OU model with t0=0,
x0=r= −70,s2=2 andS= −60, and withq=5 (bottom)
andq=10 (top).
posed for time-homogeneous diffusion processes and mentioned in the previous subsection, in a previous paper (cf. Di Nardo et al., 1998b) it has been proved that the FPT density of a Gauss – Markov process can be obtained by solving the simple, non-singular, Volterra second kind inte-gral Eq. (5), with S(t)C1([t0, )) and
By making use of this result, in Di Nardo et al. (1998b) an efficient numerical procedure based on a repeated Simpson’s rule has been proposed to evaluate FPT densities of Gauss – Markov processes.
In the following, we present a special non-sta-tionary Gauss – Markov neuronal model and make use of such numerical procedure to analyze the corresponding firing pdf’s.
3. A special Gauss – Markov neuronal model
In this section, we consider the neuronal model {X(t), tI}, with I[0, +), characterized by easily seen, the above conditions characterizing Gauss – Markov processes are satisfied by Eqs.
(12) and (13). Indeed, in this case we haveh1(t)= s2qet/q
/2 and h2(t)=e−t/q. Hence, here we are
defining a Gauss – Markov neuronal model. Re-calling Eq. (11), the coefficientsA1(x,t) andA2(x, t) for the underlying process are, respectively, given by
A1(x,t)= −1
q(x−r)+le
−t/a,
A2(x,t)=s2. (14)
Hence, the infinitesimal moments of the OU neuronal model turn out to be a special case of model expressed by Eq. (14). Indeed, forl=0 Eq. (14) yields A1(x, t)= −(x−r)/q; moreover,
when l\0 and a¡0 the drift A1(x, t) goes to
−(x−r)/q.
Let us consider the deterministic model (sug-gested by Eqs. (1) and (14) in the absence of randomness, i.e. with s2=0) described by
dx(t)
Recalling Eqs. (12) and (13), we note that, similarly to the OU model, again the conditional mean M(tt0), given by the first of Eq. (10), identifies with Eq. (15). Furthermore, ift0=0 and
x0=r, from Eq. (15) one has
x(t)=r+lj(t),
that coincides with the mean m(t) given by Eq. (12).
We note that relations Eq. (14) can be a poste-riori interpreted in the following way. The neu-ron’s membrane potential is not only subject to the usual spontaneous exponential decay and to endogenous random components, but it also expe-riences an external input whose magnitude, how-ever, exponentially damps with the time-constant
Fig. 3. Firing pdf’s for the neuronal model of Section 3, for t0=0, x0=r= −70, s2=2, S= −60, q=5, l=14 and
a=1, 2, 3, 4. Decreasing modes refer to increasing values ofa.
parameters mimic the effect of an external neu-ron’s input whose initial strength l exponentially damps with the time-constant a. When x05r,
which is the interesting case in neurobiology, from Eq. (15) it follows that x(t) is initially increasing, to decrease monotonically as t towards the resting potential r, after reaching a maximum. Note the significant diversity of behavior of x(t) forl\0 and forl=0, the latter representing the deterministic version of the OU model. Indeed, if
l=0, x(t) monotonically tends to the resting potential r for all x0"r.
4. Numerical results
In this section, we discuss some features of the Gauss – Markov neuronal model presented above, on the ground of computational results obtained by making use of the numerical procedure of Di Nardo et al. (1998b).
Figs. 3 – 6 show some examples of firing pdf’s. Here, we have taken t0=0, x0=r= −70 and
s2=2, and we have presented separately the four
cases arising when the threshold is S= −60 or
S= −65, and when q=5, l=14, or q=10,
l=7. Each figure refers to various choices of a. In Figs. 3 and 4, we have chosen a=1, 2, 3, 4, while a=0.5, 1, 2, 3, 4 in Figs. 5 and 6.
In all cases, the firing pdf’s are unimodal; the magnitudes of the modes appear to be increasing in a, while their abscissae decrease with a. The latter are listed in Table 1.
The firing pdf’s in Figs. 3 – 6 should be com-pared with those appearing in Figs. 1 and 2 for the OU model specified by the same choice of the
Fig. 4. As in Fig. 3 but forq=10 andl=7.
Fig. 5. As in Fig. 3 but forS= −65 anda=0.5, 1, 2, 3, 4.
Table 1
Abscissae of the modes of the firing pdf’s plotted in Figs. 3–6
S= −60 S= −65
a
q=5; q=10; q=5; q=10;
l=7
l=14 l=14
l=7
0.5 1.56 5.95 0.50 0.99
2.56 0.42 0.86
1 1.22 0.94
2 2.11 0.38 0.77
0.74 0.37
1.89 3 0.87
0.37 1.77
0.83
4 0.72
Fig. 7. Cumulative distribution functions corresponding to the firing pdf’s presented in Fig. 3; from bottom to top it isa=1, 2, 3, 4.
ron to have fired before t are enhanced by large values of a. Hence, the firing times described by the random variable T are stochastically ordered in a decreasing fashion (see, for instance, Shaked and Shanthikumar, 1994) as agrows larger. Such a behavior is expected on the grounds of the drift monotonicity for the Gauss – Markov model; in-deed, from Eq. (14) it follows that A1(x, t) is increasing in a. On the contrary, it could be shown that the firing times corresponding to dif-ferentavalues do not satisfy the stronger hazard-rate order criterion.
5. Conclusions
Use of a recently developed computational method is made in order to compare the firing probability densities and distributions for the clas-sical OU model and for a generalization of it, which is obtained by assuming the existence of an external exponentially damped input. The per-formed analysis pinpoints the quantitative changes of the neuron’s output firing features, that is characterized — as expected — by an overall firing enhancement.
Acknowledgements
This work has been performed within a joint cooperation agreement between Japan Science and Technology Corporation (JST) and Univer-sita` di Napoli ‘Federico II’, under partial support by CNR and by MURST.
Fig. 8. Cumulative distribution functions corresponding to the firing pdf’s presented in Fig. 4; from bottom to top it isa=1, 2, 3, 4.
Fig. 9. Cumulative distribution functions corresponding to the firing pdf’s presented in Fig. 5; from bottom to top it is
a=0.5, 1, 2, 3, 4.
Fig. 10. Cumulative distribution functions corresponding to the firing pdf’s presented in Fig. 6; from bottom to top it is
a=0.5, 1, 2, 3, 4.
neu-References
Buonocore, A., Nobile, A.G., Ricciardi, L.M., 1987. A new integral equation for the evaluation of first passage time probability densities. Adv. Appl. Prob. 19, 784 – 800. Di Nardo, E., Nobile, A.G., Pirozzi, E., Ricciardi, L.M.,
1998a. On a non-Markov neuronal model and its approxi-mations. Biosystems 48, 29 – 35.
Di Nardo, E., Nobile, A.G., Pirozzi, E., Ricciardi, L.M., 1998b. A computational approach to first-passage-time problems for Gauss – Markov processes. Submitted for publication.
Gerstein, G.L., Mandelbrot, B., 1964. Random walk models for the spike activity of a single neuron. Biophys. J. 4, 41 – 68.
Giorno, V., Nobile, A.G., Ricciardi, L.M., Sato, S., 1989. On the evaluation of first-passage-time probability densities via nonsingular integral equations. Adv. Appl. Prob. 21, 20 – 36.
Karlin, S., Taylor, H.M., 1981. A Second Course in Stochastic
Processes. Academic Press, New York.
Kostyukov, A.I., 1978. Curve-crossing problem for Gaussian stochastic processes and its application to neural modeling. Biol. Cybern. 29, 187 – 191.
Kostyukov, A.I., Ivanov, Y.N., Kryzhanovsky, M.V., 1981. Probability of neuronal spike initiation as a curve-crossing problem for Gaussian stochastic processes. Biol. Cybern. 39, 157 – 163.
Ricciardi, L.M., 1976. Diffusion approximation for a multi-in-put model neuron. Biol. Cybern. 24, 237 – 240.
Ricciardi, L.M., 1995. Diffusion models of neuron activity. In: Arbib, M.A. (Ed.), The Handbook of Brain Theory and Neural Networks. The MIT Press, Cambridge, pp. 299 – 304.
Ricciardi, L.M., Di Crescenzo, A., Giorno, V., Nobile, A.G., 1999. An outline of theoretical and algorithmic approaches to first passage time problems with applications to biologi-cal modeling. Math. Japonica 50, 247 – 322.
Shaked, M., Shanthikumar, J.G., 1994. Stochastic Orders and Their Applications. Academic Press, San Diego.
