Probabilistic Principles for Biophysics and Neuroscience: Entropy Production, Bayesian Mechanics & the Free-Energy Principle

Abstract

This thesis focuses on three fundamental aspects of biological systems; namely, entropy production, Bayesian mechanics, and the free-energy principle. The contributions are threefold: 1) We compute the entropy production for a greater class of systems than before, including almost any stationary diffusion process, such as degenerate diffusions where the driving noise does not act on all coordinates of the system. Importantly, this class of systems encompasses Markovian approximations of stochastic differential equations driven by colored noise, which is significant since biological systems at the macro- and meso-scale are generally subject to colored fluctuations. 2) We develop a Bayesian mechanics for biological and physical entities that interact with their environment in which we give sufficient and necessary conditions for the internal states of something to infer its external states, consistently with variational Bayesian inference in statistics and theoretical neuroscience. 3) We refine the constraints on Bayesian mechanics to obtain a description that is more specific to biological systems, called the free-energy principle. This says that active and internal states of biological systems unfold as minimising a quantity known as free energy. The mathematical foundation to the free-energy principle, presented here, unlocks a first principles approach to modeling and simulating behavior in neurobiology and artificial intelligence, by minimising free energy given a generative model of external and sensory states.

Figures (26)

• Figure 1: Helmholtz decomposition. The upper left panel illustrates the Helmholtz decomposition of the drift into time-reversible and time-irreversible parts: the time-reversible part of the drift flows towards the peak of the stationary density, while the time-irreversible part flows along its contours. The upper right panel shows a sample trajectory of a two-dimensional diffusion process stationary at a Gaussian distribution. The lower panels plot sample paths of the time-reversible (lower left) and time-irreversible (lower right) parts of the dynamic. Purely conservative dynamics (lower right) are reminiscent of the trajectories of massive bodies (e.g., planets) whose random fluctuations are negligible, as in Newtonian mechanics. Together, the lower panels illustrate time-irreversibility: If we were to reverse time, the trajectories of the time-reversible process would be statistically identical, while the trajectories of the time-irreversible process be distinguishable by flow, say, clockwise instead of counterclockwise. The full process (upper right) is a combination of both time-reversible and time-irreversible dynamics. The time-irreversible part defines a non-equilibrium steady-state and induces its characteristic wandering, cyclic behaviour.
• Figure 2: Entropy production as a function of time-irreversible drift. This figure illustrates the behaviour of sample paths and the entropy production rate as one scales the irreversible drift $b_{\mathrm{irr}}$ by a parameter $\theta$. The underlying process is a two-dimensional Ornstein-Uhlenbeck process, for which exact sample paths and entropy production rate are available (Section \ref{['sec: OU process']}). The heat map represents the density of the associated Gaussian steady-state. One sees that a non-zero irreversible drift induces circular, wandering behaviour around the contours of the steady-state, characteristic of a non-equilibrium steady-state (top right and bottom left). This is accentuated by increasing the strength of the irreversible drift. The entropy production rate measures the amount of irreversibility of the stationary process. It grows quadratically as a function of the irreversible scaling factor $\theta$ (bottom right). When there is no irreversibility (top left), we witness an equilibrium steady-state. This is characterised by a vanishing entropy production (bottom right).
• Figure 3: Exact simulation of linear diffusion process with $b_{\mathrm{irr}}(x) \in \operatorname{Rang e}\sigma$. This figure considers an OU process in 3d space driven by degenerate noise, i.e., $\operatorname{ran k} \sigma <3$. The coefficients are such that $\sigma=Q$ are of rank $2$. In particular, $b_{\mathrm{irr}}(x) \in \operatorname{Rang e}\sigma$ holds for every $x$. The process is not elliptic nor hypoelliptic, but it is elliptic over the subspace in which it evolves. The upper-left panel shows a sample trajectory starting from $x_0=(1,1,1)$. The upper-right panel shows samples from different trajectories after a time-step $\varepsilon$. There are only two principal components to this point cloud as the process evolves on a two dimensional subspace. In the bottom panel, we verify the theoretically predicted value of $e_p$ by evaluating the entropy production of an exact simulation $e_p(\varepsilon)$ with time-step $\varepsilon$. As predicted, we recover the true $e_p$ in the infinitesimal limit as the time-step of the exact simulation tends to zero $\varepsilon \to 0$. Furthermore, since the process is elliptic in its subspace, the entropy production is finite.
• Figure 4: Exact simulation of linear diffusion process with $b_{\mathrm{irr}}(x) \not\in \operatorname{Rang e}\sigma$. This figure considers an OU process in 3d space driven by degenerate noise. The coefficients are such that $\operatorname{Rang e} b_{\mathrm{irr}}$ is two-dimensional while $\operatorname{Rang e}\sigma$ is one-dimensional, and such that the process does not satisfy Hörmander's hypoellipticity condition. As such the process is hypoelliptic on a two-dimensional subspace; see a sample trajectory in the upper-left panel. By hypoellipticity its transition kernels are equivalent in the sense of measures, although far removed: On the upper right panel we show samples from different trajectories after a time-step $\varepsilon$. There are only two principal components to this data-cloud as the process evolves on a two dimensional subspace. In the bottom panel, we verify the theoretically predicted $e_p$ by evaluating the entropy production of an exact simulation $e_p(\varepsilon)$ with time-step $\varepsilon$. As predicted, we recover $e_p=+\infty$ in the infinitessimal limit as the time-step of the exact simulation tends to zero $\varepsilon \downarrow 0$. This turns out to be as the transition kernels of the forward and time-reversed processes become more and more mutually singular as the time-step decreases.
• Figure 5: Exact simulation of underdamped Langevin dynamics. This figure plots underdamped Langevin dynamics in a quadratic potential. Here, the process is two dimensional, i.e., positions and momenta evolve on the real line. We exploit the fact that underdamped Langevin in a quadratic potential is an Ornstein-Uhlenbeck process to simulate sample paths exactly. The choice of parameters was: $V(q)=q^2/2, M=\gamma =1$. The upper left panel plots a sample trajectory. One observes that the process is hypoelliptic: it is not confined to a prespecified region of space, cf. Figures \ref{['fig: OU process b in Im sigma']}, \ref{['fig: OU process b not in Im sigma']}, even though random fluctuations affect the momenta only. The upper right panel plots samples of the forward and time-reversed processes after a time-step of $\varepsilon$. In the bottom panel, we verify the theoretically predicted $e_p$ by evaluating the entropy production of an exact simulation $e_p(\varepsilon)$ with time-step $\varepsilon$. As predicted, we recover $e_p=+\infty$ in the infinitessimal limit as the time-step of the exact simulation tends to zero $\varepsilon \downarrow 0$. This turns out to be because the transition kernels of the forward and time-reversed processes become more and more mutually singular as the time-step decreases.

Theorems & Definitions (49)

• Definition 2.3.1: Time reversed process
• Definition 2.3.2: Path space measure
• Remark 2.3.3: Cadlag Markov processes
• Definition 2.3.4: Restriction to a sub-interval of time
• Theorem 2.3.5
• Definition 2.3.6: Entropy production rate of a stationary Markov process
• Remark 2.3.7: Physical relevance of Definition \ref{['def: epr']}
• Proposition 2.3.8
• Proposition 2.3.10: $e_p$ in terms of transition kernels
• Definition 2.3.11