Closed-form modeling of neuronal spike train statistics using multivariate Hawkes cumulants Nicolas Privault

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Closed-form modeling of neuronal spike train

statistics using multivariate Hawkes cumulants

Nicolas Privault∗

Division of Mathematical Sciences

Nanyang Technological University

21 Nanyang Link, Singapore 637371

Michèle Thieullen†

LPSM-UMR 8001 - Case Courrier 158

Sorbonne Université

4 Place Jussieu, 75252 Paris Cedex 05, France

October 28, 2022

Abstract

We derive exact analytical expressions for the cumulants of any orders of neuronal

membrane potentials driven by spike trains in a multivariate Hawkes process model

with excitation and inhibition. Such expressions can be used for the prediction and

sensitivity analysis of the statistical behavior of the model over time, and to estimate

the probability densities of neuronal membrane potentials using Gram-Charlier expan-

sions. Our results are shown to provide a better alternative to Monte Carlo estimates

via stochastic simulations, and computer codes based on combinatorial recursions are

included.

Key words: Multivariate Hawkes processes; ﬁltered shot noise processes; multivariate

cumulants; Gram-Charlier expansions; excitatory synapses; inhibitory synapses; membrane

potentials.

1 Introduction

Hawkes processes [Haw71] are self-exciting point processes that have been applied to the

modeling of random spike trains in neuroscience in e.g. [CR10], [KRS10], [GDT17], [CXVK19].

Neuronal spike train activity has been modeled using multivariate Hawkes processes in e.g.

[RBRTM13], [OJSBB17], [KR20], where ﬁltered Hawkes processes have been interpreted as

free membrane potentials in the linear-nonlinear cascade model. In this framework, the cu-

mulants of multivariate Hawkes processes yield important statistical information. However,

∗nprivault@ntu.edu.sg

†michele.thieullen@sorbonne-universite.fr

arXiv:2210.15549v1 [q-bio.NC] 27 Oct 2022

the analysis of statistical properties of Hawkes processes is made diﬃcult by their recur-

sive nature, in particular, computing the cumulants of Hawkes processes involves technical

diﬃculties due to the inﬁnite recursions involved.

Neuronal synaptic input has also been modeled using multiplicative Poisson shot noise

driven by random current spikes, in e.g. [VD74], [Tuc88], see also [KAR04], [RD05], [RG05],

[Bur06a], for the analysis of stationary limits in the case of constant Poisson arrival rates,

and [WL08,WL10], see also [AI01], [Bur06b], [CTRM06] for time-dependent Poisson in-

tensities modeling of time-inhomogeneous synaptic input. In this framework, the time

evolution of the probability density functions of membrane potentials has been described

in [BD15], [Pri20] by Gram-Charlier probability density expansions based on moment and

cumulant estimates.

The computation of the moments of Hawkes processes has been the object of several

approaches, see [DZ11], [CHY20] and [DP22] for the use of diﬀerential equations, and

[BDM12] for stochastic calculus methods applied to ﬁrst and second order moments. Other

techniques have been introduced for linear and nonlinear self-exciting processes, including

Feynman diagrams [OJSBB17], path integrals [KR20], and tree-based methods [JHR15]

applied up to third order cumulants. However, such methods appear diﬃcult to implement

systematically for higher order cumulants, and they use ﬁnite order expansions that only

approximate cumulants even in the linear case.

In this paper, we provide a recursion for the closed-form computation of the cumulants

of multivariate Hawkes processes, without involving approximations. For this, we extend

the recursive algorithm of [Pri21] to the computation of joint cumulants of all orders of

multivariate Hawkes processes. This algorithm, based on a recursive relation for the Prob-

ability Generating Function (PGFl) of self exciting point processes started from a single

point, relies on sums over partitions and Bell polynomials. In what follows, we will apply

this algorithm to Hawkes processes with inhibition, by using negative weights in their clus-

ter point process construction. We note that although our cumulant expressions are proved

only for non-negative weights, the results remain numerically accurate and consistent with

the sampled cumulants of Hawkes processes with inhibition as long as the process does not

become inactive over long time intervals, see also § 1 of [OJSBB17].

In Proposition 2.1 and Corollary 2.2 we compute the joint cumulants of membrane

potentials modeled according to a ﬁltered Hawkes process as in [OJSBB17]. In comparison

with Monte Carlo simulation estimates, explicit expressions allow for immediate numerical

evaluations over multiple ranges of parameters, whereas Monte Carlo estimations can be

slow to implement. In addition, such expressions are suitable for algebraic manipulations

and tabulation, e.g. they can be diﬀerentiated in closed form with respect to time to yield

the dynamics of cumulants, or with respect to any system parameter to yield sensitivity

measures. Numerical applications of our closed form expressions are presented in Section 3,

where they are compared to Monte Carlo estimates. Although our simulations in Figures 2

to 5have been run with 10 million samples, Monte Carlo estimates of higher-order cumulants

can be subject to numerical instabilities not observed with closed-form expressions. In

particular, they become degraded starting with joint third cumulants (see Figure 4-b)) and

fourth cumulants (see Figure 5-a)), and they become clearly insuﬃcient for the estimation

of fourth joint cumulants (see Figure 5-b)).

Closed-form cumulant expressions are then applied in Section 4to the explicit derivation

of cumulant-based Gram-Charlier expansions for the probability density function of the

membrane potentials at any given time. showing that densities are negatively skewed with

positive excess kurtosis.

We proceed as follows. In Section 2we present closed-form recursions for the com-

putation of cumulants of any order in a multivariate Hawkes process model. Numerical

results are then presented in Section 3with application to the modeling of connectivity in

spike train statistics. In Section 4we present numerical experiments based on cumulants

for the estimation of probability densities of potentials by Gram-Charlier expansions. In

the appendices we present the derivation of recursive cumulant and moment identities for

the closed-form computation of the moments of Hawkes processes, in the multivariate case,

with the corresponding codes written in Maple and Mathematica.

2 Cumulants of multivariate Hawkes processes

This section describes our algorithm for the computation of cumulants. Let (H1(t), . . . , Hn(t))t≥0

denote a multivariate linear Hawkes point process with self-exciting stochastic intensities

of the form

λi(t) := νi(t) +

j=1 Zt

γi,j(t−s)dHj(s), t ∈IR+,(2.1)

with Poisson oﬀspring intensities γi,j(dx) = γi,j(x)dx and possibly time inhomogeneous

Poisson baseline intensities νi(dt) = νi(t)dt,i= 1, . . . , m. The next proposition provides

a way to compute the joint cumulants of random sums by an induction relation based on

the Bell polynomials. In what follows, we assume that γ1(IR+) + ··· +γm(IR+)<1, and

consider the integral operator Γdeﬁned as

(Γf)(x, i) =

j=1 Z∞

f(x+y, j)γi,j (dy), x ∈IR+, i = 1, . . . , m,

and, letting Idenote identity, the inverse operator (I−Γ)−1given by

((I−Γ)−1f)(x, i) = f(x, i) +

∞

n=1

(Γnf)(x, i)

=f(x, i) +

∞

n=1

j1,...,jn=1 Z∞

0···Z∞

f(x+y1+··· +yn, jn)γi,j1(dy1)···γjn−1,jn(dyn),

x∈IR+,i= 1, . . . , m. The following statements hold for the joint cumulants κ(n)

(x,i)(f1, . . . , fn)

of Pm

j=1 R∞

0fi(t, j)dHj(t),...,Pm

j=1 R∞

0fn(t, j)dHj(t)given that the multidimensional

Hawkes process is started from a single jump located in Hi(t)at time x∈IR+,i= 1, . . . , m.

Proposition 2.1 a) The ﬁrst cumulant κ(1)

(x,i)(f)of Pm

j=1 R∞

0f(t, j)dHj(t)is given by

κ(1)

(x,i)(f) = ((I−Γ)−1f)(x, i)

=f(x, i) +

∞

n=1

j1,...,jn=1 Z∞

0···Z∞

f(x+y1+··· +yn, jn)γi,j1(dy1)···γjn−1,jn(dyn),

x≥0,i= 1, . . . , m.

b) For n≥2, the joint cumulants κ(n)

(x,i)(f1, . . . , fn)are given by the induction relation

κ(n)

(x,i)(f1, . . . , fn) =

k=2 X

π1∪···∪πk={1,...,n} (I−Γ)−1Γ

j=1

κ(|πj|)

(·,·)((fl)l∈πj)!(x, i),(2.2)

x≥0,i= 1, . . . , m,n≥2, where the above sum is over set partitions (π1, . . . , πk)of

{1, . . . , n}and |πi|denotes the cardinality of the set πi,i= 1, . . . , k.

Proof. See Appendix A.

Standard (i.e. unconditional) cumulants can then be obtained in the next corollary as a

consequence of Proposition 2.1.

Corollary 2.2 The joint cumulants κ(n)(f1, . . . , fn)of Pm

j=1 R∞

0fi(t, j)dHj(t)1≤i≤nare

given by the relation

κ(n)(f1, . . . , fn) =

i=1

k=1 X

π1∪···∪πk={1,...,n}Z∞

j=1

κ(|πj|)

(x,i)((fi)i∈πj)νi(x)dx, n ≥1.(2.3)

Proof. See Appendix A.

Exponential kernels

Joint cumulants will be computed using sums over partitions and Bell polynomials in the

case of the exponential oﬀspring intensities

γi,j(dx) = wi,j1[0,∞)(x)e−bxdx, i, j = 1, . . . , m,

given by the m×mconnectivity matrix W= (wi,j)1≤i,j≤m,|wi,j |< b, and the constant

Poisson intensities νi(dz) = νidz,νi>0,i, j = 1, . . . , m. In this case, the integral operator

Γsatisﬁes

(Γf)(x, i) =

j=1

wi,j Z∞

f(x+y, j)e−bydy, x ∈IR+, i = 1, . . . , n.

The recursive calculation of joint cumulants can be performed using the family of functions

ep,η,t,j(x, i) := 1{i=j}xpeηx1[0,t](x),η < b,p≥0, by evaluating (I−Γ)−1Γin Proposition 2.1

on the family of functions ep,η,t,j as in the next lemma.

Lemma 2.3 For fin the linear span generated by the functions ep,η,t,k,p≥0,η < b,

k= 1, . . . , m, the operator (I−Γ)−1Γis given by

((I−Γ)−1Γf)(x, i) =

j=1 Zt−x

f(x+y, j)WeyW i,j e−bydy, x ∈[0, t], i = 1, . . . , m.

For fas in Lemma 2.3, by Proposition 2.1 the ﬁrst cumulant of Z∞

fi(t, j)dHj(t)given

that the multidimensional Hawkes process is started from a single jump located in Hi(t)at

time x∈IR+,i= 1, . . . , m, is given by

κ(1)

(x,i)f(·)1{j}=f(x)1{i=j}+Zt−x

e−byf(x+y)WeyW i,j dy,

x∈[0, t],i= 1, . . . , m, and for n≥2we have the recursion

κ(n)

(x,i)(f1[0,t]) =

k=2

j=1 Zt−x

e−byWeyW i,j Bn,kκ(1)

(x+y,j)(f), . . . , κ(n−k+1)

(x+y,j)(f)dy.

The conditional multivariate joint cumulants of R∞

0fi(t, ji)dHji(t)1≤i≤nare given by

κ(n)

(x,i)f11[0,t1]1{j1}, . . . , fn1[0,tn]1{jn}

j=1

k=2 X

π1∪···∪πk={1,...,n}Zmin(t1,...,tm)−x

e−byWeyW i,j

l=1

κ(|πl|)

(x+y,j)fp1{jp}p∈πldy,

j1, . . . , jn≥1, with, for n= 2,

κ(2)

(x,i)f11[0,t1]1{j1}, f21[0,t2]1{j2}=

j=1Zmin(t1,t2)−x

e−byWeyW i,j κ(1)

(x+y,j)f11{j1}κ(1)

(x+y,j)f21{j2}dy.

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Closed-formmodelingofneuronalspiketrainstatisticsusingmultivariateHawkescumulantsNicolasPrivault*DivisionofMathematicalSciencesNanyangTechnologicalUniversity21NanyangLink,Singapore637371MichèleThieullenLPSM-UMR8001-CaseCourrier158SorbonneUniversité4PlaceJussieu,75252ParisCedex05,FranceOctober28,202...

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Closed-form modeling of neuronal spike train statistics using multivariate Hawkes cumulants Nicolas Privault

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