Ecient implicit solvers for models of neuronal networks Luca Bonaventura1

2025-05-03 0 0 1.36MB 31 页 10玖币
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Efficient implicit solvers
for models of neuronal networks
Luca Bonaventura(1),
Soledad Fern´andez-Garc´ıa(2),(3), Macarena G´omez-M´armol(2)
October 5, 2022
(1) Dipartimento di Matematica
Politecnico di Milano
Via Bonardi 9, 20133 Milano, Italy
luca.bonaventura@polimi.it
(2) Departamento de Ecuaciones Diferenciales y An´alisis Num´erico,
Universidad de Sevilla
Apdo. de correos 1160, 41080 Sevilla, Spain
soledad@us.es, macarena@us.es
(3) Instituto de Matem´aticas de la Universidad de Sevilla,
Universidad de Sevilla
Av. de la Reina Mercedes, s/n, 41012 Sevilla, Spain
Keywords: Implicit methods, DIRK methods, biological neural networks,
slow-fast dynamics.
AMS Subject Classification: 34K28, 37M05, 65P99, 65Z05, 92C42
1
arXiv:2210.01697v1 [math.NA] 4 Oct 2022
Abstract
We introduce economical versions of standard implicit ODE solvers
that are specifically tailored for the efficient and accurate simulation
of neural networks. The specific versions of the ODE solvers proposed
here, allow to achieve a significant increase in the efficiency of network
simulations, by reducing the size of the algebraic system being solved
at each time step, a technique inspired by very successful semi-implicit
approaches in computational fluid dynamics and structural mechan-
ics. While we focus here specifically on Explicit first step, Diagonally
Implicit Runge Kutta methods (ESDIRK), similar simplifications can
also be applied to any implicit ODE solver. In order to demonstrate
the capabilities of the proposed methods, we consider networks based
on three different single cell models with slow-fast dynamics, including
the classical FitzHugh-Nagumo model, a Intracellular Calcium Con-
centration model and the Hindmarsh-Rose model. Numerical experi-
ments on the simulation of networks of increasing size based on these
models demonstrate the increased efficiency of the proposed methods.
2
1 Introduction
Synchronization between neuronal activities plays an important
role in the understanding of the nervous system. Starting with the
seminal work of Hodgkin-Huxley [18], the complexity of the ionic dy-
namics is usually reflected in the models of neural activity by the
presence of nonlinearities and of different timescales for the different
variables. The rich variety of synchronization types that can take place
in neuron networks results both from the complexity of the neural dy-
namics and from the scale and structure of the network itself, which
can vary from a small number of cells (microscopic scale), through
neuron populations (mesoscopic scale), to large areas of the brain and
spinal cord (macroscopic scale).
The synchronization properties of coupled systems with multiple
timescales, such as relaxation oscillators [15, 29], bursters [17] and sys-
tems presenting Mixed-Mode Oscillations (MMOs) [10], differ strongly
from those between coupled harmonic oscillators, and the role of the
coupling strength is different in the two cases. In particular, it has
been shown that canard phenomena [3] arising in multiple timescale
systems play a prominent role in organizing the synchronization of
coupled slow-fast systems. Recent studies on these topic mostly ad-
dress issues such as synchronization and desynchronization, local os-
cillations and clustering [11] and the problem of synchronization of
coupled multiple timescale systems constitutes a very active field of
research. Furthermore, neuronal networks with similar properties are
the main component of many supervised learning methods based on
recurrent neural networks, such as continuous time Liquid State Ma-
chines, see e.g., [28, 27] and Echo State Networks [22, 31].
Due to the nonlinearities involved and to the scale of the networks
under study, numerical simulation is an essential tool to understand
and simulate synchronization phenomena. From a numerical point of
view, efficient simulations of neural networks require the use of special
methods suitable for stiff problems, due to the slow-fast nature of the
dynamics. Furthermore, if the number of cells in the cluster is large,
numerical simulations can entail a substantial computational effort if
standard ODE solvers are applied. Numerical techniques with similar
properties are also required in the so-called Neural ODE approaches
to neural network modelling [9].
In this work, we show how to build economical versions of stan-
dard implicit ODE solvers specifically tailored for the efficient and
accurate simulation of neural networks. The specific versions of the
ODE solvers proposed here allow to achieve a significant increase in
the efficiency of network simulations, by reducing the size of the al-
3
gebraic system being solved at each time step. This development is
inspired by very successful semi-implicit approaches in computational
fluid dynamics, see e.g. [4, 8]. A similar approach was applied in
[6] to the classical equations of structural mechanics. While we fo-
cus here specifically on Explicit first step, Diagonally Implicit Runge
Kutta methods (ESDIRK), see the reviews [23, 24], analogous simpli-
fications can be applied to any implicit ODE solver.
In order to demonstrate the capabilities of the proposed methods,
we consider networks based on three different single cells models, aim-
ing to cover a wide variety of models, with different dynamical prop-
erties depending, among others, on their slow-fast nature. The first
model is the classical FitzHugh-Nagumo (FN) system [15, 29]. It con-
sists on a system of two equations that evolve with different time
scales. With only two equations, the system is able to reproduce the
neuron excitability. The second model is a FN system with an extra
variable representing the Intracellular Calcium Concentration (ICC)
in neurons [25]. The third variable is slow, so that the resulting sys-
tem is a two slow - one fast system. This allows the system to have
MMOs, that is, oscillatory patterns with an alternation of small and
large amplitude oscillations [10]. The third model is the Hindmarsh-
Rose (HR) system [17]. This is a system with one slow and two
fast variables, which displays bursting oscillations. The main char-
acteristic of these oscillations is an alternation of slow phases, where
the system is quasi-stationary, and rapid phases, where the system is
quasi-periodic. During the latter phase, the system solutions display
groups of large-amplitude oscillations or spikes that occur on a faster
timescale [21, 30]. For classical parameter values, the HR model pro-
duces square-wave bursting, one of the three main classes introduced
in [30], taking this name because of the form of the oscillations.
From each of these single-cell models, following the approaches
proposed in the literature [7, 12, 14, 20, 25, 32], we build three different
networks, based on the density of the coupling matrix (sparse, middle,
dense), with the objective of testing the efficiency of the developed
methods in these three different situations.
The rest of the article is outlined as follows: in Section 2, we
present the single neuron models that we use as node in the networks.
In Section 3, we build the networks that we aim to simulate. Subse-
quently, Section 4 is devoted to deriving the efficient implicit solvers
for the Implicit Euler method adapted to each network. After that,
in Section 5 we extend the procedure outlined in Section 4 for the im-
plicit Euler method to a class of convenient high order ODE solvers.
Numerical experiments are performed in Section 6. Finally, Section
7 is devoted to exposing conclusions and perspectives of the present
4
work.
2 Single neuron models
We present here the single-cell neuron models that we use to con-
struct the networks in Section 3. As we have already commented in
the Introduction, we consider three different slow-fast models: the
classical FN system [15, 29], the FN system with an extra variable
representing the ICC in neurons [25] and the HR system [17]. We
consider first the classical FN model of spike generation [15, 29], given
by
˙x=−y+f(x),
˙y=ε(x+g(y)),
where
f(w)=4w−w3, g(w) = a1w+a2,(1)
with (x, y)∈R2.Here xrepresents the membrane potential and y
is the slow recovery variable. The timescale separation parameter ε
fulfills 0 < ε 1,and a1, a2∈Rare usually taken such that the
system has only one equilibrium point.
We consider then the ICC model, introduced in [25], which is given
by
˙x=τ(−y+f(x)−φf(z)),
˙y=τεk(x+g(y)),
˙z=τε (φr(x) + r(z)) ,
(2)
where functions fand gare given in (1),
φf(w) = µw
w+z0
, φr(w) = λ
1 + exp (−ρ(w−xon)) ,(3)
r(w) = −w−zb
τz
,(4)
and (x, y, z)∈R3. Here, xrepresents the membrane potential, yis the
slow recovery variable and zstands for the ICC. The timescale sepa-
ration parameter εfulfills 0 < ε 1.The parameter τ > 0 has been
introduced in [25] so that the outputs comply with a given physical
timescale and does not impact the phase portrait. Moreover, following
[14], we assume a1<0,|a1|  1 and parameters a2, z0, λ, τz, zb, k to
be strictly positive. We also consider a large enough ρvalue, so that
the sigmoid φris steep at its inflection point and represents a sharp
activation function.
5
摘要：

EcientimplicitsolversformodelsofneuronalnetworksLucaBonaventura(1),SoledadFernandez-Garca(2);(3),MacarenaGomez-Marmol(2)October5,2022(1)DipartimentodiMatematicaPolitecnicodiMilanoViaBonardi9,20133Milano,Italyluca.bonaventura@polimi.it(2)DepartamentodeEcuacionesDiferencialesyAnalisisNumerico,...

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