Co-Simulation Multiscale Approach

• Co-simulation multiscale approach is a modeling framework that integrates microscale neuronal dynamics and macroscale network diffusion to capture complex brain functions.
• It employs a modified Hodgkin–Huxley model for neuron-level activity and graph-based diffusion for regional interactions, ensuring both precision and tractability.
• Incorporation of deep brain stimulation and stochastic noise facilitates realistic simulations that inform personalized therapeutic strategies for Parkinson’s disease.

A co-simulation multiscale approach refers to computational schemes in which different physical, spatial, or temporal scales are coupled via separate models or numerical solvers, exchanging information to capture hierarchical system dynamics with improved computational tractability and fidelity. Within neuroscience and neurodegenerative disorders, such multiscale modeling is imperative for elucidating how microscopic (cellular, synaptic, molecular) processes translate into macroscopic (regional, whole-brain) functional states. In the context of Parkinson’s disease (PD), a co-simulation multiscale model facilitates the integrated simulation of neuron-level electrodiffusion, population-level network interactions, and external modulatory interventions such as deep brain stimulation (DBS), incorporating additional neurobiological variability through stochastic effects (Herrera et al., 9 Sep 2025).

1. Definition and Structure of the Multiscale Co-Simulation Model

The multiscale co-simulation model for PD developed by the authors consists of two tightly coupled layers:

• Microscale (Neuron-level): The dynamics of individual neurons are governed by a modified Hodgkin–Huxley (HH) framework extended to incorporate synaptic conductance, recurrent circuitry, and stochastic synaptic input. The full set of equations for each neuron $i$ includes:

$$C\,\frac{dV_i}{dt} = g_{\mathrm{Na}}\,m_i^3 h_i\,(E_{\mathrm{Na}} - V_i) + g_K\,n_i^4\,(E_K - V_i) + g_L\,(E_L - V_i) + g_E\,(E_E - V_i) + g_I\,(E_I - V_i)$$

where $g_{E}$ and $g_{I}$ describe excitatory and inhibitory conductances updated by Poissonian synaptic events and by presynaptic network activity; $m_i$, $h_i$, and $n_i$ are the standard gating variables.

• Macroscale (Network-level): The activity across brain regions implicated in PD (specifically, cortex, basal ganglia nuclei, and thalamus) is described by a graph-based diffusion process. The anatomical connectivity is encoded via a weighted adjacency matrix $W$, leading to a graph Laplacian:

$$L_{ij} = p\,(D_{ij} - w_{ij}), \quad D_{ii} = \sum_{j=1}^V w_{ij}$$

Regional voltages then evolve under the influence of both local ionic currents and diffusive input, effectively embedding the microscale computed outputs into network propagation equations. Each region's voltage equation takes the form:

$$\frac{dv_p}{dt} = -d_y^p + \frac{\mathrm{ionic~currents} + I_{\mathrm{network},p}}{C_m}$$

2. Coupling Between Scales: Methodology and Mathematical Formulation

The integration of micro- and macroscales occurs via several explicit mechanisms:

• Each brain region contains a representative ensemble of neurons (e.g., 50 per region), each governed by the HH-type model. The population-averaged outputs (excitatory and inhibitory synaptic currents) form the inputs for that region's macroscale voltage.
• Network-level diffusion is captured by the Laplacian $L_{ij}$, which uses structural connectivity derived from diffusion tensor imaging and tractography to mediate electrodiffusive coupling between regions (subthalamic nucleus, globus pallidus, thalamus, cortex).
• The local regional activity can feed back into single-neuron models, providing realistic, context-sensitive presynaptic drive.
• Temporal integration: Both scales are simulated in a time-marched, synchronous fashion so that mesoscale phenomena (such as oscillatory synchrony, bursts, network-wide spiking) are emergent properties rather than being imposed or pre-specified.

3. Modeling of Deep Brain Stimulation (DBS) and Stochasticity

Two important aspects of the PD model are the simulation of external intervention (DBS) and the inclusion of stochastic noise, both at the single-neuron and network levels.

• Deep Brain Stimulation (DBS): DBS is modeled by incorporating a periodic stimulation current into the voltage equations of the STN and GPi neurons:

$$I_{\mathrm{DBS}} = i_p \, H\left(\sin\left(\frac{2\pi t}{P_{\mathrm{D}}}\right)\right) \left[1 - H\left(\sin\left(\frac{2\pi (t+\delta)}{P_{\mathrm{D}}}\right)\right)\right]$$

where $i_p$ is the stimulation amplitude, $P_{\mathrm{D}}$ is the stimulation period, $\delta$ the impulse width, and $H(\cdot)$ the Heaviside function. This induces high-frequency perturbations that transiently modulate the firing patterns and synchrony of affected regions, enabling direct analysis of DBS's network-level therapeutic effects.

• Stochastic Noise: Additive or multiplicative white noise is injected into the membrane voltage equations:

$$dv_i = (\text{deterministic terms})\, dt + \sigma_i\, dW_i(t)$$

where $dW_i(t)$ is a Wiener process with noise amplitude $\sigma_i$. This stochastic input emulates biological variability intrinsic to neural tissue and synaptic transmission. Analysis of simulation outputs indicates that such noise does not establish spiking activity, but instead increases the amplitude and temporal variability of regional electrical fluctuations, consistent with experimental electrophysiology.

4. Digital Brain Network Construction and Parameterization

• Graph Construction and Weighting: The adjacency matrix $W$ is parameterized using mean fiber counts and fiber length data from diffusion tractography, ensuring anatomical fidelity. The degree matrix $D$ and Laplacian $L$ provide an explicit link between structural connectivity and dynamical diffusion coefficients $p$.
• Microscale Population Sampling: Each region’s neuronal population is sampled such that the ensemble-averaged neuronal output (computed from HH equations) defines the electrical drive at the network node. This approach maintains correspondence between neuron-level phenomena (bursting, adaptation, refractoriness) and emergent mesoscopic behaviors.
• Presynaptic and Network Inputs: Synaptic conductances $g_E$ and $g_I$ in the HH model are dynamically updated both by stochastic synaptic events (modeled by Poisson statistics) and by deterministic, recurrent network activity, allowing the model to capture transitions between irregular and synchronous states (as observed in pathological PD circuits).

5. Implications for Parkinson’s Disease Dynamics and Treatment

• Mechanistic Insights: The unified model reveals that aberrant spiking and oscillatory patterns observed in the thalamus are attributable not solely to increased neural noise, but rather to pathologically strong and variable presynaptic currents driven by upstream STN and GPi activity. The addition of noise modulates, but does not generate, these patterns.
• Effect of DBS: The application of DBS—whether targeted to the STN or GPi—alters network synchrony and suppresses pathological firing, restoring more physiologically normal oscillatory activity. Model results suggest the efficacy of multiple possible stimulation sites, informing the clinical selection and tuning of DBS modalities.
• Noise and Variability: Neural noise, while not driving activity in isolation, is critical for reproducing the observed variability in biological data. This outcome supports the use of stochastic elements in future clinical modeling and simulation pipelines aimed at personalization of DBS and pharmacological therapies.
• Personalization and Extension: The scalable graph-based framework enables straightforward incorporation of individual patient connectomes or the extension to additional brain regions, supporting personalized medicine and adaptive neuromodulation strategies.

6. Technical Summary Table

Scale	Model Type	Key Variables/Eqs
Microscale	Modified HH neurons	$V_i$, $g_E$, $g_I$, noise $\sigma$
Macroscale	Network diffusion (graph)	$v_p$, Laplacian $L_{ij}$, $I_{\mathrm{network},p}$
Intervention	DBS input	$I_{\mathrm{DBS}}(t)$

The integration is realized by mapping averaged microscale variables as inputs to macroscale nodes, and reciprocally distributing the macroscale contextual information to neuron-level simulations.

7. Outlook and Future Directions

This co-simulation multiscale framework offers a computationally efficient and physiologically faithful means to simulate emergent neurodegenerative phenomena and evaluate neuromodulatory therapies. Potential extensions include:

• Incorporation of additional brain regions for whole-brain modeling.
• Dynamic learning rules for synaptic and network plasticity.
• Closed-loop simulation studies that directly inform adaptive DBS controller design.

The approach is positioned to enhance translational neuroscience by providing a theoretical and computational bridge from cell to network to system level dysfunction and treatment in PD and related disorders (Herrera et al., 9 Sep 2025).