Physics-informed Koopman learning approach for fixed-time synchronization of stochastic neural dynamics

Abstract

This study proposes a physics-informed Koopman operator framework for the fixed-time (FXT) synchronization of Hindmarsh-Rose (HR) neurons subject to mixed Gaussian and Poisson jump noise. This paper introduces the Koopman operator to lift the nonlinear neuronal dynamics into a higher dimensional space where the evolution becomes globally linear. Distinct to the data-driven methods which rely on genric basis functions, this paper design a physics-informed lifting dictionary explicitly derived from the governing algebraic nonlinearities of the HR dynamics. Based on the proposed lifted framework, this study designs a non-singular FXT controller which guarantees the convergence within a pre-determined settling time, independent of the initial error states. Further, the communication bandwidth limitations are handled through a dynamic event-triggering mechanism (DETM) ensuring the control actions are executed only when the event-triggering condition is satisfied. The numerical simulations are performed to validate the performance of the proposed theoretical frameworks.

Introduction

In the year 1952, authors Hodgkin and Huxley developed the first differential model to understand the neuronal activities that bridges the mathematics and biology¹. The proposed model explains the movement of sodium ( and potassium ( ions across the membrane. Later on, the simplified model comprises of two-dimensional model developed by FitzHugh-Nagumo (FHN) in the year 1961 which can capture the basic neuronal behaviors like periodic spiking, excitability, and refractory period². The main limitation of FHN model is considerably capturing the bursting behavioral characteristics. Hindmarsh and Rose added one more variable in the study, which is slow adaptation variable, relaxes the limitation of the FHN model³. The common factor among all the models is nonlinear characteristics and mathematically analyzing the nonlinear models are considerably challenging. The conventional methods for analyzing those characteristics are briefly discussed by the authors in^4,5. The dynamical investigation is performed for the HR by transforming the model into an equivalent linear representation^6–8. Although the eigen value based method is simple and computationally efficient, it’s approximated locally around a small neighborhood of equilibrium point of the system, which limit the accuracy, failing to capture the global nonlinear dynamics. In contrast, fuzzy based method approximates to larger range by using multiple local models, but this technique results in higher computational cost. A promising alternative viewpoint is offered by the Koopman operator theory of dynamical systems, where linear representation of the system can be achievable even for severely nonlinear dynamics through the infinite-dimensional. The theory is named after the mathematician Bernard Koopman, who introduced the operator in 1931, that explains how measurements of Hamiltonian systems change over time⁹. In 1932, the theory was extended by Bernard Koopman and Von Neumann to encompass systems having a continuous eigenvalue spectrum¹⁰. In the recent decades the studies shows tremendous advancements due to the combination of contemporary Koopman theory and a straightforward and efficient numerical realization^11–13. The modern Koopman operator theory for dynamical systems discussed by the studies offers a fundamentally different perspective by transforming nonlinear dynamics into an infinite-dimensional linear representation¹⁴ and¹⁵. This method states that observable functions of nonlinear systems can evolve linearly under the operation of a linear but infinite-dimensional operator, even when state trajectories of those systems evolve nonlinearly. The authors shows the effectiveness of Koopman operator-based control critically depends on the selection of proper lifting functions form the observable dictionary¹⁶. While generic polynomial expansions or radial basis functions can be employed, physics-informed lifting functions that exploit domain knowledge about the underlying system structure often yield superior performance with lower-dimensional representations^17,18.

Neuronal models may operate in noisy environments where stochastic fluctuations, including Gaussian and Poisson jump noises arise from multiple sources across different temporal and spatial scales. In such environments, continuous control methods consume more communication energy. Under ideal scenarios traditional control methods can achieve stability, however those methods will struggle in the presence of stochastic disturbances which eventually leads the system to slow convergence, persistent oscillations, or even loss of synchronization^19,20. To mitigate these limitations, various control strategies have been developed^21–23. Among those, FXT control strategy has the advantage of achieving convergence within a predetermined time bound, irrespective of initial conditions. Despite the aforementioned benefits, FXT control requires high communication and computational costs due to frequent updates. Event-triggered control resolves this issue by reducing the unnecessary control updates, it will update only when the prescribed triggering conditions are satisfied. By integrating FXT with event-triggered mechanisms, FXT event-triggered control achieves uniform convergence guaranties while significantly reducing communication and control effort^22,24. By ensuring the convergence of the error system, synchronization can be achieved. To theoretically guarantee the convergence without obtaining an explicit analytical solution, Lyapunov stability theory can be employed²⁵. Using Lyapunov stability theory, the FXT event-triggered control framework provides theoretical assurance and practical implementability to achieve rapid synchronization.

A key gap exists in developing the physics informed Koopman lifting method to analyze the HR neuron model by obtaining exact linear representation, without relying on data-driven learning. This study presents a fixed-time synchronization method for the HR neuron model using Koopman operator and the contributions are given below,

• This study designs and implements a FXT control strategy for the stochastic HR neuron model, which guarantees synchronization convergence of error model within a pre-determined time bound, independent of initial conditions.

• This study proposes a physics-informed Koopman operator based method by creating a dictionary of observable functions based on the nonlinearities of HR neuron model. This methodology allows the application of linear control method in the higher dimensional space while preserving the FXT convergence.

• To address the computational cost, DETM is introduced along with the FXT control. This mechanism dynamically adjust the control updates based on the convergence error, significantly reducing the computational cost without affecting the synchronization stability.

• The study analyses the system performance under Gaussian and Poisson jump noises, the synchronization quality, triggering frequency, and control energy consumption, providing insights into the practical robustness of Koopman based method.

This article structured as follows, section 1 discusses the preliminaries required for this study. Section 2 employs the stochastic HR neuron model along with the Koopman operator framework. Section 3 discusses the main results and the corollary for deterministic HR neuron model. Section 4 demonstrates the numerical results to validate the proposed framework.

Table 1.

Existing recent literature works and proposed work.

This study	HR model	Koopman	Control	Stochastic	FXT	Levy’s Noise
5
26
27
28
29
30
31
32

Preliminaries

Definition

¹⁴ Consider a nonlinear system,

1

where is the state vector.

The Koopman framework expresses nonlinear dynamics linearly in terms of observables as

2

where the space of observables is acted upon by an infinite-dimensional linear operator .

Lemma 1

^33,34 The general nonlinear model with stochastic disturbances and Levy noise is given as follows,

3

Here is a state vector. If there exist a Lyapunov function and infinitesimal operator, then the stochastic derivative is as follows,

4

Lemma 2

³⁵ For the system (2), suppose the Lyapunov function and satisfies

5

Then the settling time is obtained as follows,

6

where

Lemma 3

³⁶ For all scalar , then the following inequalities given below are satisfied,

7

where, are a scalars, with and .

Assumption 1

³⁷ If there exist a positive scalars , , , and any function , then the following inequality is satisfied.

8

Assumption 2

³³ For all where and . The Levy noise term,

9

Model formulation

The neuronal bursting and spiking were studied through a differential model proposed by Hindmarsh-Rose³, which is a three-dimensional model comprised of membrane potential and recovery variable , and slow adaptation ,

10

where the parameters a and b controls the scale of cubic and quadratic term of membrane potential respectively. The excitability of neuron is controlled by the external current . The parameter c is related to the recovery variable and d modulates how recovery variable is responding to membrane potential. The parameter r in the slow variable is the reason for time scale separation and g modulates the strength of slow adaptation current. The is the threshold value which modulating the baseline membrane potential. Numerous dynamic behaviors, including chaotic bursting, periodic oscillations, and steady states, are known to be present in the HR model.

For conciseness, the compact vector notation of the HR neuron model is given below as,

where

For this study we modify the system to incorporate stochastic disturbance g(x(t)) and Levy noise h(x(t)), then the modified master model is given as,

11

where the stochastic disturbance and Levy noise functions are defined as,

Correspondingly, the slave model is modeled with an external control input to the master model given in the equation (11) for achieving the synchronization. The dynamics of the slave model is given by,

12

where the noise coefficients are given as,

The error model is defined as, . Then the Koopman operator is introduced to HR neuron model given in equation (10) for transforming the nonlinearity of the model to linear with the observables using lifting functions. The Koopman operator usually lift the nonlinear functions to infinite linear dimension. The HR model have polynomial nonlinearities and in order to get the finite closure (finite subspace) this study will choose observables by allowing the characteristics of model behaviour, say, physics informed lifting functions.

Definition

The HR model contains polynomial nonlinearities up to the third order. So the physics informed polynomial lifting function with the vector of observable functions as,

13

Remark 1

The proposed Koopman method lift the highly nonlinear HR neuron dynamics into a higher dimensional space without the computational cost of data-driven methods (EDMD, neural networks). The physics informed lifting function explicitly created by considering the observables, which includes nonlinearities in the system (), achieves linearization. This approach allows to use the linear control theory while keeping the chaotic dynamics of the neuronal model, eliminating the drawbacks of Jacobian based linearization.

Because the Koopman lifting function () includes the polynomial nonlinearities like (), so the control signal generates is physics-informed. It anticipates the solution trajecotry of the neuron’s behaviour. Now define the Koopman induced error in the lifted space as transpose of row vector,

14

Explicitly computing each component,

To analyze the synchronization stability the error model is derived for the state . The evolution of the error is given by,

15

Here is the input matrix in lifted space. Then a nonlinear feedback control law is designed to stabilize the error dynamics. The controller consists of a linear term and two nonlinear terms dependent on the signum function, likely ensuring finite-time convergence,

16

where are control parameters, and . The event triggering condition for the control is defined as,

17

Substituting the control law u(t) into the open-loop error dynamics yields the closed-loop system. The resulting equation describes the evolution of the error under the influence of the controller,

18

The closed-loop error model is defined as follows,

19

Main results

This section discusses the theoretical results of the study which guarantees the synchronization of Koopman based stochastic HR neuron model. The sufficient conditions are derived with the help of Lyapunov stability theory and Ito’s calculus.

Theorem 1

Let be the Koopman based linear matrix, be the scalars, P is a symmetric matrix, I is an identity matrix, and is a number of jumps. If the following conditions holds,

20

then the Koopman based HR neuronal error model is globally asymptotically stable in the mean square.

Proof

Consider the Lyapunov-Krasovskii function (LKF),

21

The considered model involves in the randomness, so the traditional derivative is ineffective. With the help of Ito’s calculus in Lemma 1 and the assumption 2, the stochastic derivative of the Lyapunov function is obtained as follows,

22

Here, can be obtained as,

23

Expanding and simplifying the expression, one can get the following

24

By using the Assumption 1, one can derive the following,

25

By substituting the above equations, the equation (25) can be simplified as follow,

26

By applying Lemma 3, one can get the following inequality,

27

Substituting the Lyapunov function , we get

28

Now, consider and . Then this can leads to the following inequality,

29

Then the derivative can be written as follows,

30

where, Taking expectations for the equation (22),

31

Here expectation of stochastic and Levy noise term, . Then by integrating the aforementioned equation (31),

32

The settling time expression can be obtained based on the Lemma 2 and the discussion by the authors in³⁸,

33

This completes the proof.

Corollary 1

Consider the HR neuronal model is deterministic. If the following conditions hold, the considered deterministic Koopman based HR error model is globally asymptotically stable.

34

Here, be the Koopman based linear matrix, K be the scalars. By applying the Lemma 2, the settling time is obtained as follows,

35

This study focuses the sufficient conditions for the both deterministic and stochastic models, which can guarantee the global stability conditions of the Koopman based HR-based error neuronal model.

Numerical simulations

Numerical simulations are performed to validate the results obtained from the proposed method, physics informed Koopman operator for FXT synchronization of stochastic HR model. The schematic representation of the proposed method is given in the Fig. 1 and the standard parameter values of HR model from the literature is given in the Table 2.The time series evolution of HR neuron model while varying the external current is shown in the Fig. 2. To understand the behaviour of attractor of HR neuron model under different values, 3D phase portrait is given in the Fig. 3.The Fig. 4 confirms synchronization of Koopman induced method. A classical synchronization method is also implemented under the same initial conditions for comparison whereas both method used nonlinear FXT control strategy with an DETM. The findings show that the proposed method requires less control effort while achieving synchronization of slave system to master system. Table 3 summarizes the quantitative performance metrics for classical method and Koopman based method. The reduction in the metrics control energy for the Koopman based method shows it uses less energy to achieve the synchronization. To evaluate the DETM the total number of triggers and communication saving were calculated. Both classical and Koopman method achieved remarkable communication savings more than when comparing to continuous time control mechanisms. But comparing to classical method, total number of triggers for Koopman based model is very less which will leads to less number of updates because of the DETM. The DETM effectively reduces the control update load. So the Koopman based stochastic HR model achieved higher savings in communication, which gives the model better synchronization in the presence of Gaussian noise and Poisson jump. According to the calculated settling time given in equation 36, the synchronization error converges to a neighborhood of zero before the settling time.

Fig. 1.

Schematic diagram of proposed physics informed Koopman induced FXT Control with DETM for HR neuron model.

Table 2.

Standard parameter values of the HR model from the literature.

Parameter	Value	Description
a	1	Controls scale of cubic term in membrane potential
b	3	Controls scale of quadratic term in membrane potential
	3.1	External current for the membrane potential
c	1	Baseline level for recovery process
d	5	Modulates how the recovery variable responds to membrane potential
r	0.006	Time scale for the slow system
g	4	Recovery variable regulator
	1.6	Threshold for spiking

Fig. 2.

Time series trajectories across three different values for the stochastic HR neuron model for (membrane potential), (recovery variable) and (slow adaptation current). The first row shows the dense firing as the increases and second row shows illustrates the recovery dynamics with respect to each spikes. The variable captures the slower adaptation mechanism that control the bursting frequency.

Fig. 3.

3D phase-space portrayal of stochastic HR neuron model for shows the evolution of system’s chaotic attractor. As the increases the attractor enlarges and occupies more volume of state space, which leads to a more complex and chaotic regime.

Fig. 4.

The plot confirms the synchronization of Koopman induced stochastic HR neuron model. Time series plots of the state variables , and for the stochastic HR model, that is master system (black), Koopman induced system (blue) and classical with nonlinear feedback control (red) with the parameters mentioned in the Table 2.

Table 3.

Performance comparison between Classical and Koopman based methods for stochastic HR neuron model.

Metric	Classical Method	Koopman Method
Control Energy (E)	5250.98	4051.05
Total Number of Triggers (N)	9,688	7,576
Average Triggering Rate (Hz)	4.84	3.79
Communication Savings (%)	95.16	96.21

As shown in the Fig. 5, for three different external current values Koopman based HR model achieving synchronization to the true system. Figure 7 shows the error response of synchronization for the Koopman based HR model with the time evolution. The error decreases rapidly and settling to zero within a few seconds confirms the FXT stability of proposed model. Irrespective the multiple initial conditions error is settling to zero after time instants. The triggering rate shown in the Fig. 6b for Koopman based HR model is lower throughout the simulation when compared to classical model. The controller needs to update often to eliminate the error because of the unpredictable and aggressive growth caused by the raw nonlinear dynamics. Because the lifted space linearises the dynamics, the error evolution is more well-behaved and predictable. The system stays within the acceptable error bound () for longer periods between updates, resulting in a lower duty cycle (Fig. 7).

Fig. 5.

Synchronization of Koopman induced stochastic HR neuron model states , and for three different values for the external current parameter I.

Fig. 6.

The time evolution on control input u(t) shows that the Koopman based control(blue) has less cost for synchronization than the classical non linear control (red) shown in figure (a). The Koopman based control maintaining the lower energy with consistency. The time-varying triggering rate of Koopman based control (blue) and classical nonlinear control (red) is shown here figure (b). The Koopman based control maintains low triggering rate throughout the interval by keeping the error within FXT bounds.

Fig. 7.

The error convergence behavior shows that the error is collapsing down near to zero within few seconds, and it’s confirms the FXT synchronization. The zoomed versions claim that the Koopman based system is synchronizing with the true system.

Algorithm 1.

Physics Informed Koopman Induced FXT Control with DETM for HR Neuron Model

Remark 2

The integration of dynamic event triggering mechanism (DETM) achieves communication savings compared to continuous-time controller. The adaptive threshold optimizing the balance between performance and energy use by avoiding minor fluctuations. The comparison metrics provided in Table 3 show that the Koopman based event-triggering control mechanism achieves a significant reduction in total triggers, which leads to substantial communication savings while maintaining synchronization stability.

36

where, ,

Remark 3

This study validates the robustness of the proposed model under stochastic enivironment which is characterised by Gaussian and Poisson jump noises. The linear classical control methods degrade under perturbations, but the Koopman based method maintains the synchronization stability by using the lifted linear representation. The implemented FXT control guarantees synchronization error within the pre-determined time ) independently of initial conditions, unlike the asymptotic or finite-control methods.

Figure 8 reveals event-triggering instants during the first 50 seconds and reveals a dense phase at (s) where both controllers update continuously to achieve the initial FXT convergence. However, a clear divergence in behavior emerges in the steady state (s). The Koopman based system (blue) exhibits significantly sparser triggering events compared to the classical system (red). This will lead to less number of updates in event-triggered control mechanism which shows the Koopman based method has less communication cost than classical method. The control energy used by both methods during the simulation period is plotted in the Fig. 6a. The time-varying triggering rate of Koopman based control (blue) and classical nonlinear control (red) is shown in the Fig. 6b, which indicates Koopman based method maintains low triggering rate throughout the simulation period by keeping the error within a small bound. It is observed that Koopman based HR model maintaining the lower energy with consistency during the entire simulation period.

Fig. 8.

Triggering instants of classical and Koopman based HR model.

Deterministic HR model

This subsection discusses the deterministic version of HR neuron model by removing the stochastic disturbance and Levy’s noise. All parameters are kept same as in the previous section except the stochastic perturbation term which are set to zero here. The time series evolution of HR neuron model under different values are shown in the Fig. 12. The 3D phase portrait 10 shows how the HR neuron model evolve from stable attractor to chaotic attractor when is increases. The dynamical behaviour of HR neuron model while varying the two parameters and r is demonstrated in the Fig. 11. The deterministic system simulated under the same numerical method and initial conditions. This allows to understand the system dynamics with the synchronization which will drive purely by the true system. The control energy and time-varying triggering rate of Koopman based control (blue) and classical nonlinear control (red) for deterministic HR neuron model is shown in the Fig. 9a and b respectively, which indicates Koopman based method consuming less control energy and maintaining low triggering rate throughout the simulation period. The performance comparison given in the Table 4 shows that the Koopman based HR model uses less control energy than classical HR model to achieve synchronization. This allows the Koopman model controller to follow the trajectory along the system’s flow. The lower triggering rate indicates that the Koopman model requires fewer updates to maintain the error within the FXT stability bounds. The synchronization of the deterministic Koopman based HR model to true system is shown in the Fig. 14. The Fig. 13 illustrates the error convergence of deterministic system to the zero irrespective of the different initial conditions. Taken together, the stochastic and deterministic simulations demonstrate that the proposed method is robust to noise while retaining strong convergence properties in the noise-free setting

Fig. 12.

This figure illustrates the time series evolution of HR neuron model for . As the values increases membrane potential () moves from stable state to dense and chaotic firing. Where the recovery variable tracks the membrane potential spikes and slow adaptation current controls the bursting frequency.

Fig. 10.

The 3D phase portraits shows the HR neuron model evolve from stable attractor to chaotic attractor as the external current varies from .

Fig. 11.

The two parameter bifurcation diagram for HR model shows the transition from stable (blue) regions to unstable (green/yellow) as the increases. Here the parameter r is varied from 0.001 to 0.01. The 3D representation of the states and over the parameters and r. The peaks corresponding to the membrane potential and recovery variable represents the region where neuron highly excitable, whereas the slow adaptation current parameter with more less peaks and shows a gradual adaptation response the parameters.

Fig. 9.

The control energy and triggering rate for Koopman method (deterministic HR model) are very less when compared to classical method (deterministic HR model).

Table 4.

Performance comparison between Classical and Koopman based control methods for deterministic HR neuron model.

Metric	Classical Method	Koopman Method
Control Energy (E)	3415.0563	2640.7486
Total Number of Triggers (N)	8941	7970
Average Triggering Rate (Hz)	4.4705	3.9850
Communication Savings (%)	95.53	96.02

Fig. 14.

The synchronization of Koopman induced (blue) HR model to the true system without the stochastic perturbations. The nonlinear feedback control (red) is also plotted here.

Fig. 13.

The error convergence of the Koopman induced HR model without the stochastic perturbations.

Both stochastic and deterministic simulation confirms the robustness of proposed method. This confirms the global linearization using Koopman theory provides a reliable synchronization of slave system to master system regardless of the environmental noises.

Conclusion

This study establishes an effective control framework for synchronization of stochastic HR neuron model by integrating physics informed Koopman method with fixed time and DETM. By constructing a lifting function using dictionary of observables directly from the non linearities in the system, the method achieves linearization which overcomes the limitations of local approximations and mitigate the computational cost. The proposed FXT control design achieves synchronization convergence of error model within a pre-determined time bound independent of initial conditions. The DETM integrated with FXT control, which helps to minimize the control updates while maintaining the synchronization error convergence. This study shows higher robustness and and less control energy than classical method under Gaussian and Poisson jump noises. In future this work can be further extended to pre-defined synchronization, also other neuronal models and time delays can also be considered.

Author contributions

M. P: Writing - review & editing, Writing - original draft, Software, Methodology, Conceptualization. P.M: Writing - review & editing, Supervision, Software, Investigation.

Funding

Open access funding provided by Vellore Institute of Technology.

Data availability

The data generated and/or analysed during the current study are available in Zenodo repository, [https://zenodo.org/records/19436926].

Declarations

Competing interests

The authors declare no competing interests.