A Formal Probabilistic Model of the Inhibitory Control Circuit in the

Brain

Elisabetta De Maria

, Benjamin Lapijover

, Thibaud L’Yvonnet

, Sabine Moisan

and Jean-Paul Rigault

Universit

e C

ote d’Azur, CNRS, I3S, Sophia Antipolis, France

INRIA, Sophia Antipolis, France

ﬁ

Keywords:

Inhibitory Control, Biological Neural Networks & LI&F Model, Probabilistic Model, Model Checking.

Abstract:

The decline of inhibitory control efﬁciency in aging subjects with neurodegenerative diseases is due to anatom-

ical and functional changes in (pre)frontal regions of the brain. Among these regions, the basal ganglia play

a central role in the inhibitory control loop. We propose a probabilistic formal model of the biological neural

network governing the inhibitory control function and we study some of its relevant dynamic properties. We

also explore how parameter variations inﬂuence the probability for the model to display some key behaviors.

We model the different structures of the inhibitory control loop thanks to discrete Markov chains representing

Leaky Integrate and Fire neurons. The model is implemented and veriﬁed using the PRISM framework. The

ﬁnal aim is to detect sources of pathological behaviors in the neural network responsible for inhibitory control.

1 INTRODUCTION

This work proposes a formal model of the neural net-

work governing the inhibitory control function of the

human brain and studies some of its dynamic prop-

erties. We chose this function because it has been

studied for a long time and several models already

exist (Verbruggen and Logan, 2009);moreover, it is

managed by a restricted amount of brain structures

(basal ganglia) which makes it easier to model than

other cognitive functions. The aim is to artiﬁcially

represent the behavior of inhibitory control in hu-

mans through a probabilistic model. The main ad-

vantage of probabilistic models is their ability to rep-

resent a wide variability of behaviour with a single

model. In the context of early onsets of neuropatholo-

gies, this approach is convenient as even healthy sub-

jects are not necessarily expected to ace clinical tests.

We model the different structures involved in the in-

hibitory control loop thanks to probabilistic discrete

Markov chains implementing a modiﬁed version of

Leaky Integrate and Fire neurons. Markov chains

are widely used in biology, especially for modeling

event driven probabilistic systems. In the case of

neurobiology, there have been various utilizations of

Markov chains to simplify neurons modeling (Saku-

mura et al., 2001; Nossenson and Messer, 2010; Inoue

et al., 2021). We rely on formal methods to implement

this model and on model checking techniques to vali-

date and explore it. More precisely, we use PRISM (a

state of the art probabilistic model checker) to imple-

ment the model and to perform model checking.

2 FORMAL MODELING AND

VERIFICATION

Model checking is a method developed in the eight-

ies (Clarke et al., 1986) for automatic veriﬁcation of

software models. It helps identify software design

problems before implementation. In our case, we use

model-checking not to verify a software tool, but to

model the functioning of a brain structure and to ex-

plore all its possible behaviors. Hence probabilistic

models are well adapted.

Among the existing probabilistic model checkers,

we chose PRISM (Kwiatkowska et al., 2011) which is

well established in the literature, and compatible with

many other tools (parameter synthesis tools or other

model checkers). PRISM is a tool for formal modeling

and analysis of systems with random or probabilis-

tic behavior. It supports several types of probabilistic

models, discrete as well as continuous.

For the modeling formalism, we rely on discrete-

time Markov chains (DTMCs), which are transition

146

De Maria, E., Lapijover, B., L’Yvonnet, T., Moisan, S. and Rigault, J.

A Formal Probabilistic Model of the Inhibitory Control Circuit in the Brain.

DOI: 10.5220/0011625700003414

In Proceedings of the 16th International Joint Conference on Biomedical Engineering Systems and Technologies (BIOSTEC 2023) - Volume 3: BIOINFORMATICS, pages 146-154

ISBN: 978-989-758-631-6; ISSN: 2184-4305

 2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)

systems augmented with probabilities. Their set of

states represents the possible conﬁgurations of the

modeled system and the transitions between states

represent the evolution of the system, which occurs

in discrete-time steps. Probabilities to transit between

states are given by discrete probability distributions.

The dynamics of DTMCs can be speciﬁed thanks

to the PCTL

∗

(Probabilistic Computation Tree Logic)

temporal logic (Hansson and Jonsson, 1994). The

main PCTL

∗

state quantiﬁers that we use in this pa-

per are X (next time), F (sometimes in the future), G

(always in the future). The until operator, U, is such

that p1 U p2 means that property p1 remains true until

property p2 becomes true.

The most important PCTL

∗

operator is P, to rea-

son about the probability of event occurrences. P is

used to replace the usual path quantiﬁers forall and

exists. A property P bound [prop] is true in a state

s if the probability that property prop holds in all the

paths from s satisﬁes the bound bound (a compari-

son operator followed by a probability value). For ex-

ample, the property P= 0.5 [X (y = 1)] holds in a

state if the probability that y = 1 is true in the next

state equals 0.5. All the above state quantiﬁers, ex-

cept X, have bounded variants, where a time bound is

imposed on the property. Furthermore, the P PRISM

operator can be used as P=? [prop] to compute the

probability for prop to occur.

PRISM also supports positive real value user-

deﬁned ”rewards” which can be seen as counters that

do not impact the number of states and transitions of

the model nor its behavior. The R operator allows

to retrieve reward values. Additional operators deal

with reward: we mainly use C (cumulative-reward).

PRISM model checking algorithms automatically val-

idate DTMCs over PCTL

∗

or reward-based proper-

ties. They compute the actual probability of some be-

havior of a model to occur. In addition, PRISM offers

the possibility to run experiments which is a ”way of

automating multiple instances of model checking” ac-

cording to its authors. This feature allows users to ob-

tain curves displaying the evaluation results of a prop-

erty with respect to one or several variables. Besides,

PRISM also proposes statistical model-checking, a

way to test properties through several simulations.

3 COGNITIVE FUNCTIONS AND

BRAIN STRUCTURES

Cognitive functions is a broad designation for brain

processes necessary in the acquisition and process-

ing of information and in reasoning (e.g., learning,

awareness, decision making (Kiely, 2014)). The con-

cept of cognitive functions is the base of many exist-

ing models of the brain mechanisms. For instance, J.

Hopﬁeld modeled the associative memory with neural

networks (Hopﬁeld, 1982). More recently, (Schmidt

et al., 2019) used artiﬁcial neuron models to under-

stand the relations between the brain waves and cog-

nitive functions such as working memory or executive

control. One of the main challenges of neurocognitive

science is to better understand the cognitive functions

and their interactions, which may improve the clinical

assessment methods and the therapies.

3.1 Biological Neuron

Biological neurons allow inter cellular communica-

tion via electrical signals. They can have thousands

of ramiﬁcations called dendrites that receive nerve

(sensory and motor) information, named afferent sig-

nals. On the other hand the axon is usually unique and

sends information (efferent signals) to other cells.

The action potential that runs through the axon is

a nerve impulse caused by the difference in ion con-

centration between the inside and the outside of the

neuron. When a neuron does not receive any sig-

nal, the membrane maintains a resting potential which

constantly balance the concentration of the potassium

ions on both sides of the membrane. Nerve impulses

modulate the membrane potential and if this poten-

tial exceeds the threshold of excitability it triggers in

turn an action potential (a.k.a. spike). A spike is a

quick rise and fall of the membrane potential. It trav-

els along the neuron axon to reach the synaptic termi-

nals (Purves et al., 2019). Depending on the emitting

neuron, the spike either directly travels to the den-

drites of the receiving neurons or triggers the release

of neurotransmitters. These neurotransmitters reach

receptors on the dendrite side of the synaptic connec-

tion. In both cases, the receiving neuron follows the

same cycle as the neuron that sent the spike.

3.2 Inhibitory Control Circuit

According to the literature (Jahanshahi et al., 2015),

the brain regions involved in the inhibitory control

cognitive function are the cortex, the basal ganglia,

and the thalamus. Together, these anatomical struc-

tures make it possible to temporarily stop the action

of the motor cortex and therefore to inhibit an irrele-

vant action initially planned. A deﬁcit in this circuit

can cause cognitive impairment (Braak et al., 2002).

The basal ganglia (ﬁgure 1) are considered as mo-

tor structures that allow movements to start and stop.

Basal ganglia are essentially a part of the motor loop

and thus of the inhibitory control circuit. They con-

A Formal Probabilistic Model of the Inhibitory Control Circuit in the Brain

147

tain several anatomical structures such as the stria-

tum (STr), composed of the putamen and caudate nu-

cleus. These two latter structures are distinct in the

circuit, but in our formal model we only represent

the STr. This simpliﬁcation is advisable to limit the

size and complexity of the model and acceptable be-

cause the striatum can be considered as a whole func-

tional zone (Johns, 2014). Basal ganglia also contain

the globus pallidus separated into two structures: the

outer segment Gpe and the inner segment Gpi (related

to the substantia nigra pars reticulata noted SNpr,

that we represent together in our model); the subtha-

lamic nucleus (STN); and the substantia nigra pars

compacta (SNpc). The cortex (Cx) and the thalamus

(Th) are also part of the inhibitory control loop but not

of the basal ganglia (Purves et al., 2019).

Figure 1: Brain coronal section of basal ganglia and its com-

ponents (from A. Gillies, M. H

aggstr

om & P. J. Lynch).

There are two pathways in the inhibitory control

loop. In the direct pathway, after receiving inputs

from the cortex, specialized STr neurons target spe-

ciﬁc neurons of the Gpi/SNpr complex with inhibitory

inputs (Mink, 1996). These inputs trigger the disinhi-

bition of the Th area, controlling the expression of the

desired motor program, leading to ”release” (Gray-

biel, 2000) the intended physical movement. In the

indirect pathway, STr neurons target the Gpe with in-

hibitory inputs leading to the dishinibition of STN. At

the same time, STN specialized neurons receive pow-

erful afferent signals from the cortex (Purves et al.,

2019) and target the Gpi/SNpr complex with excita-

tory inputs (Mink, 1996). These inputs trigger the in-

hibition of Th (Graybiel, 2000) in a way that removes

all the unwanted motor programs.

The decline of inhibitory control efﬁciency in

aging subjects is due to anatomical and functional

changes in (pre)frontal regions (Hu et al., 2014).

However, there are differences between healthy and

pathological aging. One of our goals is to differenti-

ate these two conditions. It has been shown (Crawford

et al., 2005) that in neurodegenerative diseases, such

as Parkinson’s disease, there is a degeneration of the

neurons of the substantia nigra pars compacta (SNpc).

The dopaminergic inﬂux coming from the SNpc and

targeting STr is considerably reduced. STr is then less

inhibited which consequently reduces the inhibitory

emission of the basal ganglia, removes the inhibition

of the thalamus and therefore the motor inhibition.

4 COMPUTATIONAL MODELING

4.1 Inhibitory Control Models

Several models of the inhibitory control use the

go/no-go task to study and describe its underlying

mechanisms, for example, the horse-race models re-

viewed in (Verbruggen and Logan, 2009). They see

the inhibitory control function as a competition be-

tween a ”go” and a ”stop” process in the brain. This

modeling approach gave many insights on the mech-

anisms of the inhibitory control (Schall and Godlove,

2012). With the rise of computational simulation

techniques, the past decades have seen the develop-

ment of models based on neural networks with respect

to neuroanatomy (Schroll and Hamker, 2013).

The model presented in this work is inspired by

the work of (Wei and Wang, 2016) which attempted

to model the inhibitory control function with a set of

(heavy) biological ”Leaky Integrate and Fire” neuron

networks representing each brain structure involved

in this function.

4.2 Artiﬁcial Neural Network Models

The Leaky Integrate and Fire (LI&F) model is a good

compromise between biological ﬁdelity and compu-

tational efﬁciency for mathematical analysis (Izhike-

vich, 2004).

4.2.1 Leaky Integrate and Fire Discrete Model

According to the LI&F model, the membrane of

a neuron can be represented as an electronic cir-

cuit (Brunel and Van Rossum, 2007). In this model

the intensity of the action potential is neglected,

but the instant of its occurrence plays an important

role (Paugam-Moisy and Bohte, 2012). We consider

a discrete version of the model where the membrane

potential u at time t is deﬁned by

u(t) =







∑

i=1

· x

(t) + r · u(t − 1)

i f u(t − 1) < Tau

∑

i=1

· x

(t) otherwise

where x

(t) ∈ {0, 1} is the signal received at time t

by the neuron through its i

input synapse; w

is the

weight associated with the i

input synapse; r is the

leak factor; Tau is the excitability threshold beyond

which the neuron emits an action potential. If the

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148

membrane potential at time t exceeds the threshold,

a spike is emitted and the membrane potential is reset

to zero. The neuron output function s(t) ∈ {0, 1} is

therefore deﬁned by

s(t) = 1 i f u(t) ≥ Tau, 0 otherwise

4.2.2 Generalization to Neuron Boxes

Our goal is to model the interactions of structures

of the brain (made of thousands of neurons) while

keeping model checking tractable. As modeling each

and every neurons would make the model difﬁcult to

check, we introduce a generalization of the LI&F neu-

ron to neuron boxes. We modiﬁed the equation of the

LI&F neuron to represent several neurons, each one

producing its own output at a given instant.

Each anatomical structure of the inhibitory con-

trol circuit is represented by a box of ten neurons of

LI&F type. This number is not proportional to the ac-

tual number of neurons in the brain structures. It is

a trade off between biological accuracy and compu-

tation capacity: the model checking experiments (see

5.4) show that this simpliﬁcation is realistic enough

to represent these brain structures. Indeed, a network

of ten neurons cannot mimic the behavior of com-

plex networks with thousands of neurons. However,

it shows a ”ﬁring ratio” (number of ﬁring neurons out

of the ten ones) for each time step. Moreover, the

generalization allows to model these ten neurons with

only one entity that directly computes the ﬁring ratio

at each time step. Thus boxes makes it possible to

have a behavior relatively close to a small network of

neurons, without requiring a lot of computing power.

To take into account the difference of sizes be-

tween biological structures, the weight of each con-

nection of the model was set to a value proportional

to the weight of the same connection and the number

of neurons connecting two structures in the model of

Wei and Wang. The dynamic of a box is deﬁned by:

U(t) =

∑

i=1

· X

(t) + r ·U(t − 1) · (

N−S(t−1)

)

This formula is close to the usual LI&F one (see sec-

tion 4.2.1) except that Boolean x

is replaced by in-

teger X

ranging from 0 to 10 to mimic the possible

10 neuron inputs from another box. Another simpliﬁ-

cation is that when a biological connection exists be-

tween two boxes A and B, all neurons of box A are

considered to be connected to all neurons of box B.

Thus, the new formula does not take Boolean in-

puts but input ﬁring ratio. This simpliﬁcation led to

a new parameter, N, that represents the number of

neurons in the whole box (N = 10 in the presented

model). As neurons are far less sensitive to stim-

uli after emitting a spike due to the ”refractory peri-

ods” (Purves et al., 2019), the last term of the formula

was introduced. In this term, S(t − 1) represents the

number of neurons which discharged at the previous

time step. More precisely, S(t) is deﬁned by

S(t) =

U(t)

Tau

, where 0 ≤ S(t) ≤ N

To track the activity of a box in the model, one should

check the values of S(t) of this box in the time win-

dow of interest.

Since biological inputs from the cortex are ir-

regular, the input spikes are modeled by a Poisson

law (Heeger, 2000). To approximate this irregularity,

Wei and Wang modeled these inputs as Poisson spike

trains. Our model uses the same Poisson function to

compute the activation probabilities of the cortex and

SNpc neurons: P

(

)

−λ

, where k is the number

of cortex and SNpc neurons that send a spike.

5 MODEL AND VALIDATION

The ﬁrst task was to provide a formal model of the

main interactions between the different basal gan-

glia nuclei. We developed a PRISM model to artiﬁ-

cially mimic the behavior of human inhibitory con-

trol through a probabilistic model based on discrete-

time Markov chains. Second, we automatically tested

probabilistic temporal properties of this model thanks

to model-checking to explore potential sources of

pathological behavior in the inhibitory control circuit.

The complete code of the model and supplementary

materials can be found at https://gitlab.com/ThibLY/

inhibctrlformmodel.git.

5.1 Basal Ganglia Model Overview

Neuropsychologists and neurobiologists have theo-

rized several models of the functioning of the basal

ganglia which have already been integrated into soft-

ware systems. In (Wei and Wang, 2016) the authors

proposed a neural network model of LI&F neurons

for the functional structures of the basal ganglia. They

obtained diagrams to visualize e.g., the importance of

some speciﬁc connections in inhibitory control.

We keep the same division into functional struc-

tures. As mentioned, we mainly use model check-

ing techniques that allow the automated exploration

of each state of a model to validate or to reject a given

property. However, these methods imply to imple-

ment models with lower complexity than simulation

methods. Thus, our model follows the architecture of

Wei and Wang using LI&F neuron boxes and other

adaptations, resulting in the graph of ﬁgure 2.

A Formal Probabilistic Model of the Inhibitory Control Circuit in the Brain

149

Figure 2: Inhibitory control circuit diagram. In green: di-

rect pathway, in red: indirect one. A classic arrow corre-

sponds to an excitation, a ﬂat-tipped arrow to an inhibition.

The PRISM code for a single box of neurons fol-

lows the formula in section 4.2.2. We added condi-

tions to limit the output ﬁring ratio of boxes to val-

ues between 0 and 10 as each box represents 10 neu-

rons (see ﬁgure 3). The whole model contains all the

Figure 3: Example of PRISM code of a box of neurons re-

ceiving inputs from another box and the inputs of the model.

biological structures represented in the basal ganglia

square of ﬁgure 2 plus the thalamus. We introduced

excitatory and inhibitory connections between them

and we add connections from the cortex as inputs;

these inputs follow a Poisson law determining the

number of spikes sent to the STr and Th boxes (sec-

tion 4.2.2). Finally, we implemented an additional

Delay box between the STN and the SNpr boxes to

enforce a discrete loop and we integrated alternative

connections between the SNpc and STr to differenti-

ate a healthy brain from a brain with a degenerated

SNpc. For this we deﬁned 2 different formulas (see

PRISM code in ﬁgure 4): one takes into account both

the SNpc and the cortex inputs and the second one

only takes inputs from the cortex. Since (Cheng et al.,

2010) showed that the death of dopaminergic neu-

rons reaches 70% in the later phases of Parkinson’s

disease, we considered that a pathological brain has

a 30% probability to follow the ﬁrst formula and a

70% probability to follow the second. In a healthy

brain, dopaminergic neurons are functional so the

brain model will always follow the ﬁrst formula.

5.2 PRISM Model Implementation

The model was implemented and veriﬁed with the

PRISM framework. Each neuron box is implemented

by one PRISM module. All the neuron box mod-

ules have a common structure with two variables (for

global membrane potential and ﬁring ratio) and one

set of guards and updates.

5.2.1 Healthy Inhibitory Control Model

The implementation in PRISM of a healthy inhibitory

control contains six modules: Entry (partial repre-

sentation of cortex and SNpc), STr, GPe, STN, SNpr

(representing the Gpi/SNpr complex considered as a

single functional structure), and Th. These modules

have an associated formula for their global membrane

potential and constants for their connection weights,

except Entry that generates input spikes following a

Poisson law. Their connections follow the architec-

ture of the basal ganglia shown in ﬁgure 2. However,

we add two modules. First, the Delay module is a

simpliﬁed neuron box that sends the same amount of

spikes that it receives but at the next instant. This

supplementary module is necessary as our model is a

discrete system. It allows the SNpr to receive sig-

nals from GPe and STN simultaneously, by delay-

ing signals from STN to make them reach SNpr at

the same instant as the ones from GPe. Second, the

Inhibitor module generates no-go signals at regular

intervals (every ten counts) to simulate external in-

hibitory events. In the model, it sends stop signals

to the ST N module. As shown in ﬁgure 2, the T h

module receives the ﬁnal inhibition signal; hence, if

its output ﬁring ratio is low, it indicates a successful

inhibition.

5.2.2 Parkinsonian Inhibitory Control Model

To model the behavior observed in Parkinson’s dis-

ease, we proposed an alternative formula for the

global membrane potential of STr (see ﬁgure 4). The

STr neuron box module has one more update in its set

of guards and updates. It has a probability of 0.3 to

update its membrane potential with the ”healthy” for-

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150

Figure 4: Code excerpt: brain with Parkinson syndrome.

mula and of 0.7 to update it with the ”pathological”

one, according to (Cheng et al., 2010).

5.3 Properties of Individual Boxes and

of Box Synchronization

The ﬁrst step was to validate the boxes individu-

ally. Each neuron box must respect the speciﬁcations

stated in section 4.2.2. In particular, at the box level,

the model must verify the conditions concerning the

global membrane potential and the ﬁring ratio: the

maximum and minimum global membrane potential

must not be exceeded; the maximum and minimum

number of spikes must be respected; as long as the

global membrane potential is not greater than or equal

to a threshold, there should be no spike. Thus, we ver-

iﬁed that the corresponding PCTL

∗

properties, which

are invariants of the boxes, hold for all boxes. In the

property below, X denotes the current box under test,

n X is the number of spikes emitted by box X, and

potential X is the global membrane potential of X.

As a reminder, P =? is the PCTL operator to com-

pute a probability and F = nprop indicates whether

property prop is true at time step n.

Property 1. What is the probability for the number

of spikes to always be 0 until the potential equals or

exceeds tau = 80?

P =?[n X = 0 U potential X ≥ 80]

The resting potential of a neuron is −10mV and

the biological threshold is 70mV; since the resting po-

tential in our model is 0 we chose a threshold value of

80. The PRISM model checker gives a positive answer

(P=1) for all properties and all boxes.

The next step is to validate the synchronization

of the boxes which must be connected to respect the

known properties about the connections of the corre-

sponding biological structures, as shown in ﬁgure 2.

As examples, the next properties check both the direct

and indirect pathways from the instant the stop signal

is sent (arbitrarily chosen to be the tenth).

Property 2. What is the probability for the stop

signal to rise at the 10

instant?

P =?[F = 10 n

ST N > 3]

Property 3. What is the probability for GPe and the

Delay box to be activated at the 11

instant?

P =?[F = 11 n GPe > 3 & n Delay > 3]

Property 4. What is the probability for STr to be

inhibited and SNpr to be activated at the 12

instant?

P =?[F = 12 n STr < 5 & n SN pr > 3]

PRISM explicit model checking engine gives the

expected valid answers (P=1). The model shows the

disinhibition and activation of Gpi/SNpr nuclei that

trigger the inhibition of the thalamus and thus the in-

hibition of an action. Moreover, the simulation graph

in ﬁgure 5 (obtained with the ”run experiment” and

statistical model-checking tools of PRISM) showing

spiking activity also conﬁrms the inhibition of STr

and Th following a stop signal. Though they are not at

the same scale level, these activities can approximate

the simulation results of (Wei and Wang, 2016).

Figure 5: STN-SNpr-Th path. The stop signal reaches the

STN, enables the SNpr activation (through the STr inhibi-

tion at the same time (t = 13)), and ﬁnally the thalamus

inhibition, and therefore of the behavioral response; in red:

STN, in blue: SNr, in green: Th.

5.4 Property on Thalamus Inhibition

To deduce the probability that a movement is actually

inhibited, we deﬁned a property evaluating the

probability to observe a low number of ﬁring neurons

in the thalamus box. We arbitrarily consider that the

Th box is inhibited when it releases less than 4 spikes

(40% of the maximal ”ﬁring ratio” which is 10 in our

implementation). In PCTL* this property is written:

A Formal Probabilistic Model of the Inhibitory Control Circuit in the Brain

151

Property 5. What is the probability for Th to be

inhibited at the ﬁrst stop trial (13

time-step)?

P =?[F = 13(n T h < 4)]

We compared our model checking results with the

simulation results of Wei and Wang. The behavior of

our formal inhibitory control model is close to the re-

sults presented in (Wei and Wang, 2016). This behav-

ior is still described as a race in which the stop signal

information has to transit from STN to SNpr before

the STr inputs inhibit SNpr (Schmidt et al., 2013).

Wei and Wang conducted an experiment on their

model by modulating the network weights. The goal

was to determine if the inhibitory control behavior

is more sensitive to the modulation of some connec-

tions than to others. To reproduce this experiment on

our model, we doubled the weight of all the connec-

tions one at a time and we computed the probability

to reach a state where the number of spikes emitted

by the thalamus is less than 4. The results, shown in

ﬁgure 6, go slightly beyond those of (Wei and Wang,

2016). In this ﬁgure a + indicates that increasing the

Figure 6: Inhibitory control circuit with modiﬁcation of

connection weights.

weight of the associated connection increases the Stop

Signal Reaction Time (SSRT) (lower probability for

Th inhibition), and a − indicates that increasing the

weight decreases SSRT (higher probability for Th in-

hibition). This experiment highlights two categories

of connections: those which facilitate thalamus inhi-

bition and those which prevent it. The STr-GPe, STN-

SNpr, and SNpr-Th connections belong to the ﬁrst

category while GPe-GPe, STN-GPe, Gpe-SNpr, and

STr-SNpr connections belong to the second one (see

ﬁgure 6). To the best of our knowledge these speciﬁc

connections are yet to be studied in vivo. Such results

need further biological experiments for validation.

Table 1 shows results and computation times for

the veriﬁcation of property 5 for the model with-

out modiﬁcation and for some connection modiﬁca-

tions. The decrease of the number of ﬁring neurons

in the STr-GPe, STN-SNpr, and SNpr-Th connections

causes a decrease in the probability of inhibition and

Table 1: Model checking results (explicit engine) for prop-

erty 5 of the model without modiﬁcations (Original) and

after weight modiﬁcation of speciﬁc connections.

Connection Result Time (s)

Original 0.8527 0.288

Gpe-SNpr 0.5251 0.303

STr-SNpr 0.4889 0.300

STr-GPe 0.9432 0.310

STN-SNpr 1 0.267

SNpr-Th 1 0.250

consequently a decrease in behavioral inhibition. A

decrease in the inhibition probability in the model

means a longer SSRT leading to the fulﬁllment of an

unwanted action. More bibliographical research may

be necessary to evaluate the relevance of these results.

5.5 Comparison of Impaired versus

Healthy Brain Models

In a healthy brain all SNpc neurons are present. As

the number of SNpc neurons depletes in Parkinson’s

disease (Cheng et al., 2010), with about 30% neu-

rons remaining in SNpc of advanced Parkinson’s, an

unhealthy brain only has 30% probability of taking

SNpc neurons into account. Still with property 5 we

Table 2: Results of property 5 for the model without modi-

ﬁcations (Healthy) and the Parkinsonian model.

Result Time (s)

Healthy 0.8527 0.288

Parkinson 0.7651 2.42

found that there is a greater chance of getting a thala-

mus inhibition in the case of a healthy brain (table 2).

The second formula in ﬁgure 4 and its associated 30%

probability are enough to lead to smaller chances of

action inhibition (of the order of 10

−1

6 CONCLUSION AND

PERSPECTIVES

A better understanding of the mechanisms of in-

hibitory control could allow targeted treatments for

different classes of patients with dementia. This work

proposed a discrete probabilistic model of the in-

hibitory control circuit of the brain and its formal val-

idation. This model reproduces known biological be-

haviors from the literature such as the pathway race.

Our model also faithfully represents the importance

of some connections in the pathways (e.g., SNr and

STr connections). These behaviors were translated in

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PCTL properties to check the adequacy of the model.

We also ran an experiment to explore the sensitivity

of inhibitory control to the modulation of some con-

nections. The modiﬁed model complies with Parkin-

son’s disease. Further modiﬁcations to represent, e.g.,

Alzheimer’s disease are planned as future work.

Probabilistic formal models can represent a wide

variety of behaviors while enabling model checking.

To check our model with standard tools, it was neces-

sary to brought up a new generalization of the LI&F

classical neuron model to represent small networks

behavior with a single module. This work opens new

avenues for the formal modeling of cognitive func-

tions. Moreover, it has proven the feasibility of such

model exploration using only off the shelf laptops.

In the future, the model will be coupled with the

activity model of a patient playing a serious game tar-

geting the inhibitory control function. The goal is to

explore modiﬁcations in the brain neural network that

may generate a patient behavior characteristic of neu-

rocognitive disorders.

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