Population spiking and bursting in next generation neural masses with spike-frequency adaptation Alberto Ferrara1David Angulo-Garcia2Alessandro Torcini3 4 5and Simona Olmi4 5

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Population spiking and bursting in next generation neural masses

with spike-frequency adaptation

Alberto Ferrara,1David Angulo-Garcia,2Alessandro Torcini,3, 4, 5 and Simona Olmi4, 5, ∗

1Sorbonne Universit´e, INSERM, CNRS, Institut de la Vision, 75012 Paris, France

2Grupo de Modelado Computacional-Din´amica y Complejidad de Sistemas,

Instituto de Matem´aticas Aplicadas, Universidad de Cartagena,

Carrera 6 36-100, Cartagena de Indias 130001, Colombia

3Laboratoire de Physique Th´eorique et Mod´elisation, UMR 8089,

CY Cergy Paris Universit´e, CNRS, 95302 Cergy-Pontoise, France

4CNR, Consiglio Nazionale delle Ricerche, Istituto dei Sistemi Complessi,

via Madonna del Piano 10, 50019 Sesto Fiorentino, Italy

5INFN, Sezione di Firenze, via Sansone 1, 50019 Sesto Fiorentino, Italy

(Dated: October 11, 2022)

Spike-frequency adaptation (SFA) is a fundamental neuronal mechanism taking into account the

fatigue due to spike emissions and the consequent reduction of the ﬁring activity. We have studied

the eﬀect of this adaptation mechanism on the macroscopic dynamics of excitatory and inhibitory

networks of quadratic integrate-and-ﬁre (QIF) neurons coupled via exponentially decaying post-

synaptic potentials. In particular, we have studied the population activities by employing an exact

mean ﬁeld reduction, which gives rise to next generation neural mass models. This low-dimensional

reduction allows for the derivation of bifurcation diagrams and the identiﬁcation of the possible

macroscopic regimes emerging both in a single and in two identically coupled neural masses. In

single popukations SFA favours the emergence of population bursts in excitatory networks, while it

hinders tonic population spiking for inhibitory ones. The symmetric coupling of two neural masses,

in absence of adaptation, leads to the emergence of macroscopic solutions with broken symmetry

: namely, chimera-like solutions in the inhibitory case and anti-phase population spikes in the

excitatory one. The addition of SFA leads to new collective dynamical regimes exhibiting cross-

frequency coupling (CFC) among the fast synaptic time scale and the slow adaptation one, ranging

from anti-phase slow-fast nested oscillations to symmetric and asymmetric bursting phenomena.

The analysis of these CFC rhythms in the θ-γrange has revealed that a reduction of SFA leads

to an increase of the θfrequency joined to a decrease of the γone. This is analogous to what

reported experimentally for the hippocampus and the olfactory cortex of rodents under cholinergic

modulation, that is known to reduce SFA.

Keywords: neural networks, mean ﬁeld models, spike-frequency adaptation, cross-frequency coupling, popu-

lation burst, population spikes, θ-nested γoscillations

I. INTRODUCTION

Neural mass models are mean ﬁeld models developed to

mimic the dynamics of homogenous populations of neu-

rons. These models range from purely heuristic ones (as

the well-known Wilson-Cowan model [1]), to more re-

ﬁned versions obtained by considering the eigenfunction

expansion of the Fokker-Planck equation for the distribu-

tion of the membrane potentials [2,3]. However, quite re-

cently, a next generation neural mass model has been de-

rived in an exact manner for heterogeneous populations

of quadratic integrate-and-ﬁre (QIF) neurons [4]. This

new generation of neural mass models describes the dy-

namics of networks of spiking neurons in terms of macro-

scopic variables, like the population ﬁring rate and the

mean membrane potential, and it has already found var-

ious applications in many neuroscientiﬁc contexts [5–13].

Neural populations can display collective events, re-

sembling spiking or bursting dynamics, observable at the

∗corresponding author: simona.olmi@ﬁ.isc.cnr.it

single neuron level [14,15]. In particular, in this context,

tonic spiking corresponds to periodic collective oscilla-

tions (COs), while a population burst is a relaxation os-

cillation connecting a spiking regime to a silent (resting)

state [16].

Regular collective oscillations have been reported for

spiking neural populations with purely excitatory [17,18]

or inhibitory interactions [19]. The emergence of these

oscillations have been usually related to the presence of

a synaptic time scale [20] or to a delay in the spike trans-

mission [21]. Indeed, as shown in [7,22], the inclusion

of exponentially decaying synapses in inhibitory QIF net-

works is suﬃcient for the appearence of COs, correspond-

ing to limit cycles emerging via a Hopf bifurcation in the

associated neural mass formulation.

A prominent role for the emergence of population

bursts is played by spike-frequency adaptation (SFA),

a mechanism for which a neuron, subject to a constant

stimulation, gradually lowers its ﬁring rate [23]. Adapta-

tion in brain circuits is controlled by cholinergic neuro-

modulation. In particular an increase of the acetylcholine

neuromodulator released by the cholinergic nuclei leads

to a clear reduction of SFA in the Cornu Ammonis area 1

arXiv:2210.04673v1 [cond-mat.dis-nn] 10 Oct 2022

(CA1) pyramidal cells of the hippocampus [24]. Recently,

it has been shown that an excitatory next generation neu-

ral mass equipped with a mechanism of global adaptation

(speciﬁcally short-term depression or SFA) can give rise

to bursting behaviours [11].

Furthermore, cholinergic drugs are responsible for a

modiﬁcation of the neural oscillations frequency, specif-

ically of the θand γrhythms [25–27], which are among

the most common brain rhythms [28]. Speciﬁcally γos-

cillations, which have been observed in many areas of the

brain [29], have a range between '30 and 120 Hz, while θ

oscillations correspond to 4–12 Hz in rodents [30,31] and

to 1–4 Hz in humans [32]. Moreover γoscillations have

been recently cathegorized in three distinct bands for the

CA1 of the hippocampus [31]: a slow one ( '30 −50

Hz), a fast one ( '50 −90 Hz), and a so-termed εband

('90 −150 Hz). γrhythms with similar low- and high-

frequency sub-bands occur in many other brain regions

besides the hippocampus [33,34] and they are usually

modulated by slower θrhythms in the hippocampus dur-

ing locomotory actions and rapid eye movement (REM)

sleep. This modulation is an example of a more general

mechanism of cross-frequency coupling (CFC) between a

low and a high frequency rhythm, which is believed to

be functionally relevant for the brain actvity [35]: low

frequency rhythms (such as θ) usually involve broad re-

gions of the brain and are entrained to external inputs

and/or cognitive events, while high frequency oscillations

(such as γ) reﬂect local computation activity. Thus CFC

can represent an eﬀective mechanism to transfer infor-

mation across spatial and temporal scales [35,36]. The

most studied CFC mechanism is the phase-amplitude

coupling, which corresponds to the modiﬁcation of the

amplitude (or power) of γ-waves induced by the phase of

the θ-oscillations; this phenomenon is often referred as θ-

nested γoscillations [37]. Cholinergic neuromodulation,

and therefore SFA, has been shown to control θ-γphase-

amplitude coupling in freely moving rats in the medial

enthorinal cortex [38] and in the prefrontal cortex [39].

In this paper we want to compare the role played by

the spike frequency adaptation in shaping the emergent

dynamics in two simple setups: either purely inhibitory

or purely excitatory neural networks. In this regard we

consider fully coupled QIF neurons with SFA, interact-

ing via exponentially decaying post-synaptic potentials.

The spike-frequency adaptation is included in the model

via an additional collective afterhyperpolarization (AHP)

current, which temporarily hyperpolarizes the cell upon

spike emission, with a recovery time of the order of hun-

dreds of milliseconds [23,25,40]. We will ﬁrst analyze the

dynamics of an isolated population with SFA and then

extend the analysis to two simmetrically coupled popula-

tions. In the latter case we will focus on the emergence of

collective solutions (either asynchronous or characterized

by population spiking and bursting) with particular em-

phasis of the symmetric or nonsymmetric nature of the

dynamics displayed by each population [41].

The emergence of CFC among oscillations in the θand

γrange have been previously reported for next generation

neural masses: namely, for two asymmetrically coupled

inhibitory populations with diﬀerent synaptic time scales

[7], as well as for inhibitory and excitatory-inhibitory net-

works under an external θ-drive [10]. Here, we show that,

in presence of SFA, θ-γCFCs naturally emerge in ab-

sence of external forcing and for symmetrically coupled

populations. The cross-frequency coupling is due to the

presence of a fast time scale associated to the synap-

tic dynamics and a slow one relative to the adaptation.

Quite peculiarly, inhibitory interactions give rise to slow

γoscillations (30-60 Hz), while excitatory ones are asso-

ciated to fast γrhythms (60-130 Hz). Furthermore, we

will show that SFA controls the frequency of the slow and

fast rhythms as well as the entrainement among θand γ

rhythms.

This paper is organised as follows. Section II is de-

voted to the introduction of the QIF neuron and of the

studied network models, as well as to the presentation

of the corresponding neural mass models. The methods

employed to characterize the linear stability of the sta-

tionary solutions for a single and two coupled populations

are reported in sub-section II.D. In sub-section III.A the

regions of existence of population spikes and bursts are

identiﬁed for a single population with SFA together with

the bifurcation diagrams displaying the possible collec-

tive dynamical regimes. The analysis is then extended to

two symmetrically coupled excitatory or inhibitory pop-

ulations without SFA in sub-section III.B and with SFA

in III.C. The relevance and inﬂuence of SFA for θ-γCFC

for two symmetrically coupled populations is examined in

Section IV. Finally a summary and a brief discussion of

the results is reported in Section V. Appendix A summa-

rizes the methods employed to study the linear stability

of the stationary solutions for a single population.

II. MODEL AND METHODS

A. Quadratic Integrate and Fire (QIF) Neuron

As single neuron model we consider the QIF Neuron,

which represents the normal form of Hodgkin class I ex-

citable membranes [42] and it allows for exact analytic

treatements of network dynamics at the mean ﬁeld level

[4]. The membrane potential dynamical evolution for an

isolated QIF neuron is given by

τ˙

V(t) = V2(t) + η(1)

where τ= 10 ms is the membrane time constant and η

is the excitability of the neuron.

The QIF neuron exhibits two possible dynamics de-

pending on the sign of η. For negative η, the neuron is

excitable and for any initial condition V(0) <√−η, it

reaches asymptotically the resting value −√−η. How-

ever, for initial values larger than the excitability thresh-

old, V(0) >√−η, the membrane potential grows un-

bounded and a reset mechanism has to be introduced

together with a formal spike emission to mimic the spik-

ing behaviour of a neuron. As a matter of fact, whenever

V(t) reaches a threshold value Vth, the neuron delivers a

formal spike and its membrane voltage is reset to Vr; for

the QIF neuron Vth =−Vr=∞. In other words the QIF

neuron emits a spike at time tkwhenever V(t−

k)→ ∞,

and it is instantaneously reset to V(t+

k)→ −∞. For pos-

itive η, the neuron is supra-threshold and it delivers a

regular train of spikes with frequency √η/π.

B. Network models of QIF neurons

We consider a heterogeneous network of Nfully

coupled QIF neurons with spike-frequency adaptation

(SFA). The membrane potential dynamics of QIF neu-

rons can be written as

τ˙

Vi(t) = V2

i(t) + ηi+JS(t)−Ai(t) (2a)

τA˙

Ai(t) = −Ai(t) + αX

m|ti

m<t

δ(t−ti

m) (2b)

i= 1, . . . , N

τS˙

S(t) = −S(t) + 1

j=1 X

k|tj

k<t

δ(t−tj

k),(2c)

where the network dynamics is given by the evolution of

2N+ 1 degrees of freedom. Here, ηiis the excitability

of the i-th neuron and Jis the synaptic strength which

is assumed to be identical for each synapse. The sign

of Jdetermines if the pre-synaptic neuron is excitatory

(J > 0) or inhibitory (J < 0). Moreover, S(t) is the

global synaptic current accounting for all the previously

emitted spikes in the network, where tj

k< t is the spike

time emission of the k-th spike delivered by neuron j. We

assumed exponentially decaying PSPs with decay rate

τS, therefore S(t) is simply the linear super-position of

all the PSPs emitted at previous times in the whole net-

work. Since we have considered a fully coupled network,

S(t) is the same for each neuron. The adaptation vari-

able Ai(t) accounts for the decrease in the excitability

due to the activity of neuron i. Each time the neuron

emits a spike at time ti

k, the variable Aiis increased by

a quantity αand the eﬀect of the spikes is forgotten ex-

ponentially with a decay constant τA. We termed this

version of the network model µ-SFA, since it accounts

for the adaptability at a microscopic level.

However, as shown in [9], by assuming that the spike

trains received by each neuron have the same statistical

properties of the spike train emitted by a single neuron

(apart obvious rescaling related to the size), the SFA can

be included in the model also in a mesoscopic way. In this

case the evolution equations for the membrane potentials

read as

τ˙

Vi(t) = V2

i(t) + ηi+JS(t)−A(t) (3a)

i= 1, . . . , N

τA˙

A(t) = −A(t) + α

j=1 X

k|tj

k<t

δ(t−tj

k) (3b)

τS˙

S(t) = −S(t) + 1

j=1 X

k|tj

k<t

δ(t−tj

k),(3c)

where the number of ODEs describing the network dy-

namics is now reduced to N+ 2 and the SFA dynamics

is common to all neurons and driven by the population

ﬁring rate

r(t) = 1

j=1 X

k|tj

k<t

δ(t−tj

k).(4)

In the following we will denote this network model as m-

SFA. The treatment of the Nadaptability variables {Ai}

in terms of a single mesoscopic one is clearly justiﬁed

i) for relatively narrow distributions of the excitabilities

with respect to the median excitability value [13,43] and

ii) for suﬃciently long adaptive time scale τA>> τS.

While the former assumption implies limiting the vari-

ability of the ﬁring rates of the single neurons, the latter

allows us to neglect the modulations of the ﬁring rates

on synaptic time scales [11].

In large part of the paper, we will consider adimen-

sional time units, therefore the variables entering in (2)

and (3) will be rescaled as follows

t=t

τ˜

S=τS, ˜

Ai=τAi,˜

A=τA (5)

and the time scales as

τs=τS

τ, τa=τA

τ.(6)

Only in the ﬁnal Secs. IV and V, we will come back

to dimensional time units in order to make easier the

comparison with experimental ﬁndings. To simplify the

notation and without lack of clarity we will omit in the

following the ( ˜ ) symbol on the adimensional variables

and parameters.

C. Neural mass models

1. Single neural population

As shown in [4], a neural mass model describing the

macroscopic evolution of a fully coupled heterogeneous

QIF spiking network with instantaneous synapses can be

derived analytically, by assuming that the excitabilities

{ηi}follow a Lorentzian distribution

g(η) = 1

∆

(η−¯η)2+ ∆2,(7)

where ¯ηis the median value of the distribution and ∆

is the half-width at half-maximum (HWHM), account-

ing for the dispersion of the distribution. The deriva-

tion is possible for the QIF neuronal model, since its

dynamical evolution can be rewritten in terms of purely

sinusoidal functions of a phase variable. This allows us

to apply the Ott-Antonsen Ansatz, introduced for phase

oscillator networks [44], in the context of spiking neural

networks [45,46]. In particular, the analytic derivation

reported in [4] allows us to rewrite the network dynam-

ics in terms of only two collective variables: the popu-

lation ﬁring rate r(t) and the mean membrane potential

v(t) = PN

i=1 Vi(t)/N. The neural mass thus introduced

can be extended to include ﬁnite synaptic decays; for ex-

ponentially decaying synapses it takes the following form

[22]

˙r=∆

π+ 2rv (8a)

˙v=v2+ ¯η−(πr)2+Js (8b)

τs˙s=−s+r , (8c)

where a third variable in now present, s(t), representing

the global synaptic ﬁeld.

The inclusion of SFA in the neural mass model Eqs. (8)

is straightforward when considering the m-SFA, since, in

this context, the adaptability is described by a collective

variable. This ﬁnally leads to the following four dimen-

sional mesoscopic model for a single QIF population

˙r=∆

π+ 2rv (9a)

˙v=v2+ ¯η−(πr)2+Js −A(9b)

τs˙s=−s+r(9c)

τa˙

A=−A+αr , (9d)

which represents the exact mean ﬁeld formulation of the

QIF spiking network with m-SFA, described by the N+2

set of ODEs (3), in the limit N→ ∞. However, as we will

show thereafter, it can also capture the dynamics of the

network with µ-SFA, described by the set of ODEs (2)

and characterized by N2+ 1 variables, for limited neural

hetereogeneity (i.e. suﬃciently small ∆) and suﬃciently

long adaptive time scales τa.

2. Two symmetrically coupled neural population

The dynamics of two symmetrically coupled identical

neural masses with adaptation is described by the 8-dim

system of ODEs

˙r1,2=∆

π+ 2r1,2v1,2(10a)

˙v1,2=v2

1,2+ ¯η−(πr1,2)2+Jss1,2+Jcs2,1−A1,2

(10b)

τs˙s1,2=−s1,2+r1,2(10c)

τa˙

A1,2=−A1,2+αr1,2; (10d)

where Jsand Jcrepresent the self- and cross-coupling,

respectively.

On the basis of Eqs. (10), one cannot dis-

tinguish between the two populations, therefore

these equations are invariant under the permuta-

tion of the variables (r1, v1, s1, A1, r2, v2, s2, A2)−→

(r2, v2, s2, A2, r1, v1, s1, A1) and they admit the exis-

tence of entirely symmetric solutions (r1, v1, s1, A1)≡

(r2, v2, s2, A2).

By following [41], we analyse the stability of symmetric

solutions by transforming the original set of variables in

the following ones







rl,t

vl,t

sl,t

Al,t





=1





















±





















(11)

where we refer to (rl, vl, sl, Al) and (rt, vt, st, At) as the

longitudinal and transverse set of coordinates, respec-

tively.

In this new set of coordinates, the trajectories of

the symmetric solutions live in the invariant subspace

(rl, vl, sl, Al, rt≡0, vt≡0, st≡0, At≡0), with the lon-

gitudinal variables satisfying the following set of ODEs

˙rl=∆

π+ 2rlvl(12a)

˙vl=v2

l+ 1 −(πrl)2+ (Js+Jc)sl−Al(12b)

τs˙sl=sl−rl(12c)

τa˙

Al=Al−αrl.(12d)

D. Linear Stability Analysis

1. Stationary solutions for a single population

By following [22], we explore the stability of the sta-

tionary solutions of the single QIF population, corre-

sponding to ﬁxed point solutions (r0, v0, s0, A0) in the

neural mass formulation (9). These ﬁxed point solutions

are given by s0=r0,A0=αr0, together with the im-

plicit algebraic system

v0=−∆

2πr0

(13)

0 = v2

0+ ¯η+Jr0−π2r2

0−αr0.(14)

Notice that the whole equilibrium solution can be

parametrized in terms of r0.

The linear stability analysis of the ﬁxed point can be

performed by estimating the eigenvalues λof the associ-

ated Jacobian matrix







2v02r00 0

−2π2r02v0J−1

τs0−1

τs0

τa0 0 −1

τa.







(15)

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Populationspikingandburstinginnextgenerationneuralmasseswithspike-frequencyadaptationAlbertoFerrara,1DavidAngulo-Garcia,2AlessandroTorcini,3,4,5andSimonaOlmi4,5,1SorbonneUniversite,INSERM,CNRS,InstitutdelaVision,75012Paris,France2GrupodeModeladoComputacional-DinamicayComplejidaddeSistemas,Institu...

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Population spiking and bursting in next generation neural masses with spike-frequency adaptation Alberto Ferrara1David Angulo-Garcia2Alessandro Torcini3 4 5and Simona Olmi4 5

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