Rendering neuronal state equations compatible with the principle of stationary action

Area of Science:

• Physics
• Computational Neuroscience
• Mathematical Neuroscience

Background:

• The principle of stationary action is fundamental in physics but not directly applicable to common computational neuroscience models.
• Existing models in computational neuroscience, like Dynamic Causal Modelling (DCM), use first-order linear differential equations incompatible with Lagrangian formulations.

Purpose of the Study:

• To adapt computational neuroscience equations, specifically DCM, to be compatible with the principle of stationary action.
• To explore the implications of a Lagrangian formulation for understanding neural dynamics.

Main Methods:

• Modified the DCM neuronal state equation using a complex dependent variable and an oscillatory solution.
• Employed a Hermitian intrinsic connectivity matrix to facilitate a Lagrangian formulation.
• Utilized Bayesian model inversion for in silico validation.
• Applied the modified model to in vivo neuroimaging datasets.

Main Results:

• Demonstrated that modified DCM equations can be formulated using the principle of stationary action.
• Showed that the modified (oscillatory) model provides a more parsimonious explanation for empirical neuroimaging timeseries data.
• Successfully identified both original and modified models using in silico generated data.

Conclusions:

• The modified Lagrangian formulation offers a compatible framework for applying the principle of stationary action to computational neuroscience.
• This approach may enable the exploration of symmetries and conservation laws in neural systems.
• The modified model presents a more parsimonious explanation for certain neuroimaging data, suggesting its utility in neuroscience research.