Rendering neuronal state equations compatible with the principle of stationary action

Abstract

The principle of stationary action is a cornerstone of modern physics, providing a powerful framework for investigating dynamical systems found in classical mechanics through to quantum field theory. However, computational neuroscience, despite its heavy reliance on concepts in physics, is anomalous in this regard as its main equations of motion are not compatible with a Lagrangian formulation and hence with the principle of stationary action. Taking the Dynamic Causal Modelling (DCM) neuronal state equation as an instructive archetype of the first-order linear differential equations commonly found in computational neuroscience, we show that it is possible to make certain modifications to this equation to render it compatible with the principle of stationary action. Specifically, we show that a Lagrangian formulation of the DCM neuronal state equation is facilitated using a complex dependent variable, an oscillatory solution, and a Hermitian intrinsic connectivity matrix. We first demonstrate proof of principle by using Bayesian model inversion to show that both the original and modified models can be correctly identified via in silico data generated directly from their respective equations of motion. We then provide motivation for adopting the modified models in neuroscience by using three different types of publicly available in vivo neuroimaging datasets, together with open source MATLAB code, to show that the modified (oscillatory) model provides a more parsimonious explanation for some of these empirical timeseries. It is our hope that this work will, in combination with existing techniques, allow people to explore the symmetries and associated conservation laws within neural systems – and to exploit the computational expediency facilitated by direct variational techniques.

Neural Activity in Quarks Language: Lattice Field Theory for a Network of Real Neurons

“…The neural dynamic is expected to follow some causal evolution influenced by the prior states, i.e., a dynamical process with memory. As already argued by several authors [ 21 , 23 , 49 , 51 , 52 , 72 ], it is reasonable to assume that such dynamics can be described by a quantum evolution, so that the formalism of quantum field theory [ 21 , 23 , 38 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 51 , 52 , 53 , 54 , 55 , 56 ] can be applied: this is a harmless assumption, since classical evolution can always be retrieved as a sub-case of quantum evolution. Then, let us assume that the evolution of in can be characterized by a discrete process of interacting binary fields, or Qubits [ 53 , 54 , 55 , 56 ].…”

“…For better or worse, the situation resembles the “zoo of particle physics” prior to the introduction of the Standard Model. In this paper we introduce a lattice field theory (LFT) [ 21 , 23 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 ] that is tailored to interpret data from multisite brain–computer interfaces (BCIs) in a systematic and physically grounded way that links the microscopic parameters to the experimental observations through well-known renormalization procedures. In short, LFTs discretize the space–time into a lattice grid and are commonly used in theoretical particle physics to facilitate numerical simulations and intractable calculations.…”

“…Our starting point will be from the novel “kernel” [ 56 ] approach to LFTs, in part because of its simplicity and in part because it allows a natural connection with the theory of spin glasses [ 56 , 57 , 58 , 59 , 60 , 61 ], which has been proposed as model of neural activity [ 19 , 22 , 62 , 63 , 64 , 65 , 66 ] and for pattern storage in memory and learning [ 15 , 16 , 18 , 67 , 68 ]. We will assume that time evolution can be characterized by a discrete non-relativistic process of interacting binary fields, or Qubits [ 53 , 54 , 55 ], that from a neuroscience point of view can be interpreted as a field theoretic version of the Free Energy principle of the Bayesian Brain Theory (see for example Friston et al [ 49 , 69 , 70 ]). We fully develop the formalism and the basic principles for binary raster diagrams (although the arguments can be readily extended to any Potts-like model with multi-spin interactions).…”

“…Achieving balanced “dissatisfaction” between physics and neuroscience readers is an important goal of this paper. Indeed, although previous proposals for neural LFTs exist [ 21 , 23 , 49 , 51 , 52 , 72 ], in our opinion, none tackled the problem of linking with the actual experimental observables in a way that can also be managed by non-physicists.…”

Neural Field Continuum Limits and the Structure–Function Partitioning of Cognitive–Emotional Brain Networks

“…A set of well-known conditions, termed preferential attachment rules, emulate the Hebbian-like and nonHebbian-like synaptic or nodal plasticity rules represented in natural and artificial systems, and are thus suitable for the present purposes [ 8 , 81 , 82 , 84 , 98 , 99 ]. Preferential attachment rules of complex technological networks [ 87 , 88 , 89 , 90 ] and biological networks [ 81 , 82 , 98 , 99 ] obey classical Maxwell-Boltzmann, quantum Fermi-Dirac, and quantum Bose-Einstein statistics, which dictate continuum limits on the system comprised of classical and quantum neurons and synapses [ 100 , 101 , 102 , 103 , 104 , 105 , 106 , 107 , 108 ].…”