Computational Methods

The CPNS Lab's approach to modelling the brain: generative models, variational inference, and neural dynamics.

Overview

Our modelling framework integrates neural mass models, Bayesian inversion, and active inference to investigate psychiatric conditions and build neuro-inspired AI systems. We develop and apply methods that combine conductance-based biophysics with advanced variational inference techniques, enabling interpretable and flexible characterisation of circuit mechanisms.

Thalamo-Cortical Model (TCM)

The TCM is a conductance-based neural mass model capturing voltage-dependent ion channel dynamics across thalamic and cortical populations. It includes AMPA, NMDA, GABA-A, GABA-B, M-type, and H-type currents, with compartmental separation and transmission delays.

Example dynamics:

\[ C_m \frac{dV}{dt} = -g_{\mathrm{AMPA}}(V - E_{\mathrm{AMPA}}) - g_{\mathrm{NMDA}}(V - E_{\mathrm{NMDA}}) - g_{\mathrm{GABA_A}}(V - E_{\mathrm{GABA_A}}) + \cdots \]

Fixed point estimation: To compute the Laplace-domain transfer function, we linearise the system around a fixed point \( x^* \) using Newton-Raphson iteration. This involves solving:

\[ f(x^*) = 0 \quad \Rightarrow \quad x_{n+1} = x_n - J^{-1}(x_n) f(x_n) \]

where \( J = \frac{\partial f}{\partial x} \) is the Jacobian matrix of partial derivatives.

A full circuit diagram and implementation are available on our GitHub.

Laplace Transfer Function with Delays

We linearise the system around the fixed point and compute the transfer function in the Laplace domain, capturing the system’s frequency response:

\[ Y(s) = C (sI - A)^{-1} B U(s) + C (sI - A)^{-1} x_0 \]

Incorporating delays: To model finite transmission delays \( \tau_{ij} \) between populations, we substitute:

\[ A \rightarrow A \odot e^{-s \tau}, \]

where the Hadamard product is applied elementwise between \( A \) and the delay kernel \( e^{-s \tau} \). This modifies the transfer function to:

\[ Y(s) = C (sI - A \odot e^{-s\tau})^{-1} B U(s) \]

This approach supports accurate spectral fits in DCM using biologically realistic axonal delays.

Thermodynamic Variational Laplace (ThermoVL)

We use an extended Laplace inference scheme with smarter variance updates, low-rank + diagonal covariance structure, and thermodynamic integration to estimate posterior uncertainty and model evidence.

ELBO Objective:

\[ \mathcal{F} = \mathbb{E}_{q} [ \log p(y|\theta) ] - \text{KL}[ q(\theta) \| p(\theta) ] \]

Key Steps:

  1. Initialise prior \( p(\theta) \sim \mathcal{N}(m_0, S_0) \)
  2. Linearise model \( y \approx f(\theta) + \epsilon \)
  3. Estimate posterior \( q(\theta) = \mathcal{N}(m, S) \) using curvature \( H = \nabla^2 \log p(y|\theta) \)
  4. Represent \( S \approx VV^\top + D \) for efficiency
  5. Anneal posterior from prior to posterior with \( \beta \in [0, 1] \):
\[ \mathcal{F}(\beta) = \mathbb{E}_q [ \beta \log p(y|\theta) + \log p(\theta) - \log q(\theta) ] \]

Thermodynamic integration approximates \( \log Z \) via:

\[ \log Z = \int_0^1 \mathbb{E}_{q_\beta} [ \log p(y|\theta) ] d\beta \]

Resources & Code