Abstract
This study presents a new application of the Cyclic-Small Gain Theorem (CSGT) to analyze the behavior of a Neural Mass Model (NMM), which is utilized to study brain activity related to epilepsy. The model is reformulated as an interconnected network of dynamic subsystems, which made CSGT applicable to this context. It is shown that whenever the CSGT conditions are satisfied, the model is input-to-state stable, and epilepsy cannot occur. Furthermore, the proposed method guarantees normal activity even with increased amplitude of noise, as long as the stability conditions of CSGT are verified. In this way, system instability can be considered indicative of the occurrence of epilepsy. The behavior of an interconnected network consisting of two epileptic columns and a healthy column is analyzed using the proposed method to study the propagation of epileptic activity between regions. The results show that, when the healthy column satisfies the CSGT conditions, it remains stable and unaffected by its epileptic counterparts. The study emphasizes the importance of the CSGT as a protective measure against epileptic activity and provides insights for designing control interventions using electric brain stimulation.
Subject terms: Electrical and electronic engineering, Dynamic networks, Systems analysis
Introduction
Approximately 70 million individuals worldwide have epilepsy1, which is the second most prevalent neurological disorder on a global scale2. Efficient tactics are required to alleviate the consequences of epilepsy and improve the quality of life for individuals and the society. In this regard, mathematical modeling and analysis are cornerstones of research on this topic.
Neural mass model (NMM) provides a computational framework for studying the dynamics of large-scale neural activity in the brain. This model captures the collective behavior of populations of neurons and their interactions within complex neural networks. By simulating the interactions between excitatory and inhibitory populations of neurons, NMM offers insights into the underlying mechanisms of brain activity and can be particularly useful in understanding epileptic seizures.
The NMM, which is also well-known as Wendling model, is a neuronal dynamic network model that was introduced in3 and developed in4–6. It produces responses similar to electroencephalogram (EEG) signals, representing brain activity. The similarity between Wendling model outputs and real EEG signals, in all stages from normal to epileptic activity, makes it a suitable base model for the analytical study of brain activities related to epilepsy.
In the Wendling model, a column represents a small region of the cerebral cortex that can be healthy or the focal area (the area where epileptic activity has begun), and for the column, an aggregated dynamics of order ten is considered. Roughly, simultaneous excessive activity in the column dynamics, as a result, synchrony, leads to epileptic activity in the model output in the corresponding cortical area. The propagation of epileptic activities in different brain areas is a complex process. However, the Wendling model offers a promising way for understanding this process. Coupling several columns provides a comprehensive framework for understanding the propagation of epileptic activities.
The main quantitative feature of the occurrence of epilepsy in EEG signals is a peak in Power Spectral Density (PSD). Roughly, epileptic signals can be considered as oscillatory signal at a specific frequency, or slow quasi-sinusoidal activity. On the other hand, it is observed that an unstable system with a saturation-like function in the loop can produce oscillatory outputs. In this case, instability does not cause unbounded output signals due to the existence of saturation-like functions. In the Wendling model, there exist some sigmoidal functions that saturate their outputs. This fact is a motivation to relate the occurrence of epilepsy to internal instability and healthy behavior to the internal stability of the Wendling Model.
The Wendling model has multiple interconnected loops. Stability analysis of this model needs some advanced tools that can handle the interaction of interconnected dynamics. In the paper, the Cyclic Small Gain Theorem (CSGT), an extension of the classical small gain theorem suitable for analyzing multiple interconnected loops, is employed to analyze the input-to-state stability (ISS) of the Wendling model. To the best of the authors’ knowledge, this is the first work that relates the stability/instability to the occurrence of epilepsy and paves the way for employing the control theory literature in the analytical study of brain dynamics for epilepsy treatment.
It is worth noting that the bifurcation analysis of the Wendling model supports the results of this paper. In15, it is shown that there exists a Hopf bifurcation for some parameters of this model, and a limit cycle with an unstable equilibrium point inside it is related to epileptic output.
In this paper, a column represents a small region of the cerebral cortex as a graph-based interaction of dynamic nodes. It is then shown that epileptic output cannot be observed if that column’s CSGT conditions are satisfied. An interesting result is that columns that do not satisfy the CSGT conditions may exhibit epileptic output when the input noise is increased. In the meantime, columns that satisfy the CSGT conditions produce normal output regardless of the amount of input noise. As the final contribution reported in this paper, we consider a network of columns that represents the connections between adjacent areas in the cerebral cortex and demonstrate that if the CSGT conditions are met in the area adjacent to the focal area, epileptic activity does not propagate.
It is important to note that the physiological findings reported in9–11,14,16 support the analytical results of the proposed method, which can be used to gain valuable insights into the properties of the model dynamics and provide a solid foundation for designing robust control strategies for neural systems.
The paper is organized as follows. The Wendling model and the CSGT are introduced in the Materials and methods section. The stability analysis of both single and multiple columns is presented in the Results section. Simulations demonstrating the broad applicability of the proposed method are provided in the Simulation section. Finally, the Conclusion section discusses the findings, outlines the limitations, and suggests directions for future work.
Materials and method
Model equations
In this paper, an NMM known as the Wendling model is used to analyze brain activity dynamics. This model can be considered in the category of macroscopic models and is widely used as a based model for mathematical analysis of epileptic dynamics12–14. The output of this model produces different signals representing different brain activities during the transition from the interictal to the ictal period, which is strikingly realistic compared to EEG signals recorded from patients5. An overview of this model is depicted in Fig. 1.
Fig. 1.
Block diagram of NMM known as Wendling model.
One can identify interconnected loops related to four subpopulations of the brain neuronal populations: the primary pyramidal neurons, the excitatory interneurons, and the slow and fast inhibitory interneurons. Each subpopulation comprises a dynamic linear transfer function and a static nonlinear sigmoid function. The second-order linear transfer function converts the average firing rate into the average postsynaptic potential (PSP) for each subpopulation. Linear transfer functions are represented by their impulse responses as follows:
 |
where 
, 
, and 
are excitatory, slow inhibitory, and fast inhibitory impulse responses, respectively. Parameters A, B, and G are average synaptic gains and a, b, and g are the average membrane time constants. A nonlinear sigmoid function transforms the average PSP of the subpopulation into the average firing rate, which is represented by:
 |
where 
is the maximum firing rate of the subpopulation, 
is the value of the potential for which a 50% firing rate is achieved, and r describes the slope of the sigmoid at 
. In Fig. 1, constants 
s are the average number of synaptic contacts, and u(t) is Gaussian white noise with mean, 
Hz and variance, 
Hz, representing the influence of neighboring cortex areas.
The overall model equations can be written as follows:
 |
1 |
where 
, 
, and 
are representative of the average PSP of the pyramidal cells, excitatory interneurons, and fast inhibitory interneurons, respectively. 
and 
are representative of the slow inhibitory interneuron. A sample value of parameters for the brain’s normal activity was reported in5.
Cyclic small gain theorem
In its original form, the small gain theorem analyzes the stability of a dynamic system’s single loop. This theorem is considered the counterpart of gain margin analysis for nonlinear systems. However, an extended version of this theorem is needed to analyze multiple interconnected loops. Liu et al.8 developed a Lyapunov formulation of this theorem for multi-loop systems. They formulated the ISS of the interconnected system based on the ISS of each subsystem. They named their method the CSGT for continuous-time dynamical networks. Here, this method is reviewed briefly. It is used 
throughout the paper to denote the Euclidean norm of 
. A function 
is positive definite if 
for all 
and 
. 
is a class 
function if it is continuous, strictly increasing and 
; it is a class 
function if it is a class 
function and also satisfies 
as 
. 
is a class 
function if, for each fixed s, the mapping 
belongs to class 
with respect to r and, for each fixed r, the mapping 
is decreasing with respect to s and 
as 
. For nonlinear functions 
and 
defined on 
, inequality 
represents 
for all 
. 
represents the identity function and 
Consider the system
 |
2 |
where 
is piecewise continuous and locally Lipschitz in 
and 
. The input 
is a piecewise continuous, bounded function of 
for all 
The system (2) is said to be ISS if there exist a class 
function 
and a class 
function 
such that for any initial state 
and any bounded input u(t), the solution x(t) exists for all 
and satisfies
 |
Proposition 1
Consider Eq. (2) and suppose that there exists a continuously differentiable ISS-Lyapunov
function
satisfying
- there exist
such that:
- there exist
such that for all
and for all
condition
results in
for all
, where
is continuous positive definite function.
Then, the system (2) is ISS.
Proof
This proposition is an special case of Theorem 4.19 in7. 
It is assumed that the interconnected network to be studied contains N nodes. Let’s node set to be as 
. Node 
has the following dynamics:
 |
3 |
for 
where 
is augmented vector of all nodes states and 
.
In Eq. (3), it is assumed that 
is continuous and locally Lipschitz with respect to x locally uniformly with respect to 
for 
. The control variable 
is a measurable and locally bounded function.
Assumption 1
For every 
-subsystem (
), there exists a smooth ISS-Lyapunov function 
, satisfying:
- there exist
, 
such that 
- there exist
{0} 
and 
{0} such that for all 
and for all 
condition
results in
where
is a continuous and positive definite function.
A graph can describe the interconnection between nodes. This graph has a directed edge from 
to 
if 
. A path from 
to 
is a sequence like 
where 
, 
and all 
for 
are in edge set. A simple cycle is a path with identical starting and ending nodes and no other repeated nodes.
Proposition 2
Dynamical network (3) with Assumption
1is ISS if for each simple cycle
, the condition
 |
where
, is satisfied.
Proof
See8. 
Results
In order to use CSGT for analysis of Eq. (1), it must be reformulated as a graph-based interaction of dynamic nodes. Considering each neural subpopulation as a node, the graph is obtained as depicted in Fig. 2. Ingredients of nodes 
- 
in Fig. 1 are highlighted in brown, red, blue, green, and purple, respectively. Please note that 
and its neighbor S[v] are common in dynamics of 
and 
; we showed them in green border with a purple text color. Dynamics of each node in Fig. 2 can be described as:
Fig. 2.
Graph of the model for writing equations in the form of the CSGT. Nodes, paths, and colors correspond with Fig. 1 and Eq. (1). The u(t) is an input noise connected to the pyramidal neuron.
 |
4 |
where w represents the vector of neighbors states, 
and 
are the corresponding coefficients and 
is a positive constant. For example, for node 
, one has 
, 
, 
, and 
In order to study the ISS property of Eq. (4), the sigmoid function is considered as:
 |
is shifted sigmoid function and satisfies the condition 
. For numerical values of parameters r, 
, and 
, the following condition is satisfied for all 
:
 |
5 |
Therefore, Eq. (4) can be rewritten as:
 |
6 |
Proposition 3
System (6) with considering
as inputs is ISS with ISS-Lyapunov function
Proof
For Eq. (6) let’s define:
 |
7 |
Both eigenvalues of A equal to 
. Therefore, the following linear matrix Lyapunov equation has a unique positive definite solution P:
 |
and its solution can be calculated as:
 |
Consider
 |
as a Lyapunov function. Due to following fact
 |
8 |
it is obvious that condition 1 of Assumption 1 is satisfied. Derivative of V along the trajectories of the nonlinear system is as follows:
 |
Straightforward calculations result in:
 |
9 |
Using Holder’s inequality the following is result:
 |
10 |
Let’s define:
 |
11 |
Substituting Eq. (11) and (10) in Eq. (9) results in:
 |
12 |
For obtaining the last inequality of Eq. (12), Eq. (5) is used. Straightforward calculations show that the conditions
 |
13 |
result in
 |
Consequently, from Eq. (8) the following is result:
 |
Condition (13) is equivalent to:
 |
14 |
From Eq. (8), it is clear that if
 |
15 |
are satisfied then Eq. (14) is held. Therefore, condition 2 of Proposition 1 is satisfied with 
and 
. 
Stability of a single column
Here, the stability of Eq. (1) via Proposition 2 is studied. The graph of subsystems of Eq. (1) is depicted in Fig. 2 and dynamics of each node is in the form of Eq. (4).
For each subsystem consider the following Lyapunov function:
 |
where 
and 
is the solution of following equation:
 |
and 
is as Eq. (7) with 
corresponds the node i parameters. With these Lyapunov functions, condition 1 of Assumption 1 is satisfied due to Eq. (8).
Similar to the proof of Proposition 3, one can conclude that the conditions
 |
16 |
result in
 |
where 
, 
and 
are determined via comparison Eq. (4) and dynamics of node i. Now consider:
 |
17 |
Similarly:
 |
18 |
Therefore, whenever
 |
19 |
hold, then Eq. (16) holds, and consequently, Assumption 1 holds. In Eq. (19), 
is the corresponding 
s coefficients in Eq. (17) and Eq. (18). In this regard, the gain of each path denoted as 
, which goes from node j to node i, is calculated as:
 |
Proposition 4
Dynamical system (1) considering u(t) and bias of sigmoids as inputs is ISS if the following conditions hold:
 |
Proof
System (1) has four loops, as shown in Fig. 2. Due to the Proposition 2 and discussions in this subsection, this system is ISS if the gain of each loop is less than one. Please note that class 
functions 
in Proposition 2 are here as 
. 
Stability of multi columns
Focal epilepsy typically originates from a specific region in the brian and then propagates to adjacent or distant brain areas17. This phenomenon has been extensively studied from multiple perspectives. From a physiological standpoint, studies have demonstrated the presence of structural and anatomical that facilitates seizure spread across cortical regions18,19,25. From a mathematical and dynamical systems perspective, several works have analyzed how inter-region interactions and coupling strength affect the stability and transition to epileptic states 20,21. Additionally, numerous modeling and simulation studies have employed interconnected neural mass models to replicate and investigate the propagation mechanisms of epileptic activity 2,4,9,22.
In this subsection, the proposed method is employed to study the stability of multi columns. Beyond the occurrence of epilepsy in multi columns, propagation of epileptic activity can be analytically studied in this way. Here, a graph network connecting three columns is made, as shown in Fig. 3. However, this method can be applied straightforwardly to an arbitrary number of columns.
Fig. 3.
In the interconnected network of multicolumns, new edges emerge that represent the connection of adjacent areas in the cerebral cortex. These edges are shown in solid red. In this example, nodes of epileptic columns (on the left and right of the above figure, labeled 3 and 1, respectively) are shown in blue, and those of the healthy column (labeled 2) are in gray. In the 
, i is the column number, and j represents the node number.
In multicolumns connections, new edges emerge, as seen in Fig. 3. Consequently, these edges lead to the appearance of new loops beyond each column’s original internal loops. 
denotes connection strength that represents the effect of the output of column j on column i. As an example, effects of columns 2 and 3 on column 1 change the fourth equation of Eq. (1) as follows (See e.g.,2 and assume that the effect of delay in connections can be modeled with 
):
 |
New loops consisted of direct paths between the following nodes:
 |
According to the discussion in section “Stability of a single column” and network dynamic equations, the gain of each path from column j to column i is denoted as 
, which is obtained as :
 |
where 
and 
relate to node 1 of column i and 
relates to node 0 of column j.
Simulation
Simulations are conducted to examine column behavior under various conditions, including changing parameters, high-frequency input noise, and interconnected columns. All simulations were performed in MATLAB using the ODE45 solver, which is based on an adaptive Runge-Kutta method. An example of parameter for Normal activity are reported in Table 1. Parameters of all simulations was selected based on5
15 to categorize activity types belonging into the normal, preictal, and ictal periods.
Table 1.
Set of parameter values for producing normal activity5.
| Parameter | Description | Value |
|---|---|---|
| A | Average excitatory synaptic gain | 4 mV |
| B | Average slow inhibitory synaptic gain | 0.5 mV |
| G | Average fast inhibitory synaptic gain | 0.5 mV |
| a |
Inverse time constant in the feedback excitatory loop |
100 s
|
| b |
Inverse time constant in the slow feedback inhibitory loop |
50 s
|
| g |
Inverse time constant in the fast feedback inhibitory loop |
350 s
|
 ,
|
Average number of synaptic contacts in the excitatory feedback loop |
 ,
|
 ,
|
Average number of synaptic contacts in the slow feedback inhibitory loop |
 ,
|
 ,
|
Average number of synaptic contacts in the fast feedback inhibitory loop |
 ,
|
|
Average number of synaptic contacts between slow and fast inhibitory interneurons |
135 |
|
Mean of Gaussian input noise | 90 Hz |
|
Standard deviation of Gaussian input noise |
60 Hz |
|
Parameters of the sigmoid function |
|
Epileptic activity by changing parameters
It is demonstrated in Fig. 4a. that the model exhibits epileptic activity through parameter manipulation. Subsequently, after implementing CSGT conditions, the model produces normal EEG, as shown in Fig. 4b. Simulations are conducted, and findings about the impact of particular parameters on the occurrence of epileptic behavior are analyzed.
Fig. 4.
(a) Column output shows epileptic activity with parameters 
, 
and 
(b) Column output shows normal activity with parameters 
, 
and 
(satisfying CSGT condition).
High-frequency input noise
It is shown that increasing input noise mean and variance results in epileptic activity belonging to the ictal period15, and our simulations confirm it, as can be seen in Fig. 5a. However, the CSGT was applied to a column with the increased value of input noise, demonstrating that under CSGT conditions, increasing input frequency does not cause epileptic activity. The result is depicted in Fig. 5b.
Fig. 5.
(a) Epileptic activity caused in the model output by input noise with parameters mean = 250 Hz and variance = 350 Hz shows abnormal synchronization and activity in the ictal period. (b) The column output under CSGT conditions with input noise parameters mean = 250 Hz and variance = 350 Hz shows normal activity.
Three Interconnected columns
The behavior of a network with three interconnected columns is examined, focusing on how different types of column activity can impact each other. Fig. 6 shows that epileptic columns influence the healthy column, leading to transit from normal to epileptic activity in the healthy column. The result of applying the CSGT is shown in Fig. 7. The healthy column is unaffected by epileptic outputs and shows normal activity.
Fig. 6.
(a) Parameters of the column are 
, 
, 
, 
and 
to show healthy activity while the column is affected by the outputs of other columns and shows epileptic activity. (b, c) Outputs of the epileptic columns caused by parameters show activity in the ictal period with parameters 
, 
, 
, 
, 
, 
, 
, 
, 
and 
.
Fig. 7.
The column parameters was chosen 
, 
, 
, 
and 
to satisfy CSGT that results in normal activity in output and the column is not affected by the epileptic outputs of other columns. (b),(c) Epileptic columns output caused by parameters show activity in the ictal period with parameters 
, 
, 
, 
, 
, 
, 
, 
, 
and 
.
Conclusion
In this study, the CSGT is used to analyze the output of an NMM, specifically focusing on its implications for epileptic and normal brain activity. By considering system instability as an indication of epileptic activity, conditions are identified under which the column shows normal activity. Recent physiological findings in10
11
14
16 that state inhibiting inhibitory interneurons should be explored to investigate the possibility of seizure suppression confirm our results where we decreased parameters B and G to satisfy CSGT conditions.
Additionally, the impact of an increased value of input noise is explored, representing the increased activity of other cortical areas, on inducing epileptic activity in the column. Our findings indicate that an increased value of input noise can indeed lead to epileptic activity. However, by satisfying the conditions of the CSGT, the column remains stable and shows normal brain activity, even when there is increased noise amplitude. Furthermore, our analysis is extended to a network of three interconnected columns. Specifically, two epileptic columns are connected to a normal column, so the effect of two epileptic columns resulted in the healthy column changing its dynamic and showing epileptic activity. Remarkably, it is observed that the column adhering to the conditions of the CSGT continues to exhibit normal brain activity despite the presence of epileptic columns in the network. Our results are supported by9,21,23,24, which states that decreasing the strength of the connection between a epileptic column and a healthy one can prevent seizure propagation and suppress epileptic activity.
The findings presented in this study open up possibilities for future research and can serve as a foundation for the design of control strategies aimed at suppressing epileptic activity. Considering the effects of epilepsy treatments especially deep brain stimulation on brain dynamics, synaptic gains (both excitatory and inhibitory), and interconnection strengths between regions, a controller can be designed to adjust these parameters so that the CSGT conditions are satisfied. It is also worth noting that the proposed method is conservative by nature, as it relies on sufficient conditions to guarantee system stability. In future work, we plan to reduce this conservatism by refining the theoretical framework and extending it to accommodate larger networks with more cortical columns and more diverse interconnections. This will enhance the applicability of the approach to more complex and realistic brain models.
Author contributions
Sina Hosseini: Investigation, Writing-Original draft preparation, Simulation. Abolfazl Yaghmaei: Conceptualization, Investigation, Writing- Reviewing and Editing. Fariba Bahrami: Supervision, Validation of the proposed method. MohammadJavad Yazdanpanh: Supervision, Validation of the proposed method. All authors reviewed the manuscript.
Data availability
The authors declare that the data supporting the findings of this study are available within the paper. All simulation codes are available from the corresponding author on request.
Competing interests
The authors declare no competing interests.
Footnotes
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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Associated Data
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Data Availability Statement
The authors declare that the data supporting the findings of this study are available within the paper. All simulation codes are available from the corresponding author on request.