Stability Dynamics in cognitive systems

Abstract

We present a mathematical correspondence between information-theoretic stability and predictive-processing models of cognition.[cite:@InformationTheoreticStabilityRewardFunction] A cognitive system is treated as a probabilistic inference engine that minimizes the divergence between predicted and observed signal distributions. We show that the *stability reward*—previously defined as the rate of divergence minimization—corresponds exactly to the negative time derivative of variational free energy in hierarchical generative models. This equivalence provides a geometric interpretation of learning and affect as gradient flows on informational manifolds, unifying statistical inference and adaptive control within a single principle.

1. Introduction

Predictive-processing theory describes perception and action as dual aspects of Bayesian inference (Rao & Ballard 1999; Friston 2010). A cognitive system maintains an internal generative model $p(o_t,s_t)$ of observations $o_t$ and latent causes $s_t$; through recursive updating, it seeks to minimize surprise or free energy. In parallel, information geometry (Amari 2016; Jaynes 1957) defines signal adaptation as gradient descent on divergence between successive probability distributions. This paper unites these two frameworks by identifying the stability reward of an information-theoretic system with the free-energy descent of a predictive-processing agent.

2. Background and Definitions

2.1 Cognitive State Space

Let the cognitive state at time $t$ be described by a probability distribution over latent causes,

$$q_t(s)=p(s_t|o_{1:t}),$$

where $o_{1:t}$ denotes all observations up to $t$. The generative model specifies likelihood $p(o_t|s_t)$ and prior dynamics $p(s_t|s_{t-1})$. The manifold of all such distributions, $\mathcal{Q}=\{q_t(s)\}$, carries the Fisher–Rao metric inherited from information geometry.

2.2 Variational Free Energy

Following Friston (2010), define the variational free energy

$$F(q_t)=D_{\mathrm{KL}}(q_t(s)||p(s_t|o_t))-\ln p(o_t),$$

which bounds the negative log-evidence or surprise. Minimizing $F$ with respect to $q_t$ reduces the divergence between the system’s internal state and the true posterior.

2.3 Stability Reward in Cognitive Terms

Adapting the definition of stability from Senn (2025),

$$R_s(t)=-\frac{1}{\delta t}{D}_{\mathrm{KL}}(q_{t+\delta }||q_t),$$

measures the rate of reduction in informational divergence between consecutive belief states. A system that maximizes $R_s(t)$ maintains consistency in its inference trajectory—hence stability of belief.

3. Stability and Free-Energy Minimization

3.1 Equivalence of Objectives

The temporal change in free energy is

$${\partial }_{t}F={⟨{\dot{q}}_t,{\nabla }_{q}F⟩}_g,$$

where the inner product is defined under the Fisher–Rao metric $g$. Because ${\nabla }_{q}F={\nabla }_q{D}_{\mathrm{KL}}(q_t||p(s_t|o_t))$, the natural gradient descent

$${\dot{q}}_t=-{\mathrm{grad}}_{g}F(q_t)$$

implies

$${\partial }_{t}F=-\Vert {\mathrm{grad}}_{g}F{\Vert }_g^2\le 0.$$

Comparing with the definition of $R_s(t)$,

$$R_s(t)=-\frac{1}{\delta t}{D}_{\mathrm{KL}}(q_{t+\delta }||q_t)\approx ⟨{\dot{q}}_t,{\nabla }_q\ln q_t{⟩}_g,$$

shows that maximizing stability reward is equivalent to minimizing variational free energy. Thus, the informational stability principle and the free-energy principle describe the same gradient flow on $\mathcal{Q}$.

3.2 Energetic Interpretation

In steady state, ${\partial }_{t}F=0$ and $R_s(t)=0$, marking a local informational equilibrium. Transient increases in $F$ correspond to prediction errors, while their subsequent dissipation corresponds to stability restoration—an energetic cycle consistent with thermodynamic interpretations of inference (Crooks 1999; Friston & Ao 2012).

4. Affective Gradient and Precision

4.1 Definition

Define affect as the time derivative of stability reward:

$$A(t)={\partial }_t{R}_s(t)=-\frac{1}{\delta t}{\partial }_t{D}_{\mathrm{KL}}(q_{t+\delta }||q_t).$$

Positive $A(t)$ indicates acceleration toward stability; negative $A(t)$ indicates divergence or uncertainty amplification.

4.2 Relation to Precision

In predictive coding, the precision of a prediction error modulates the rate of update (Feldman & Friston 2010). Expressing precision ${\pi }_t=(\mathrm{Var}[{ϵ}_t]{)}^{-1}$ as a weighting on the gradient of $F$, we have

$${\dot{q}}_t=-{\pi }_t{\mathrm{grad}}_{g}F(q_t),$$

so that $A(t)$ depends on the temporal derivative of precision:

$$A(t)\propto {\partial }_t{\pi }_t.$$

Affective change thus represents the felt sensitivity of the system’s confidence in its inferences.

5. Hierarchical Stability and Attention

Cognitive systems exhibit hierarchical organization: higher levels encode slower, more abstract causes, and lower levels encode faster, sensory features (Friston 2008; Clark 2013). Let levels be indexed by $l$, each with belief state $q_t^{(l)}(s^{(l)})$. Stability reward generalizes to

$$R_s^{(l)}(t)=-\frac{1}{\delta t}{D}_{\mathrm{KL}}(q_{t+\delta }^{(l)}||q_t^{(l)}),$$

and total stability is the weighted sum

$$R_s^{\mathrm{total}}=\sum _l{w}_l{R}_s^{(l)}(t),$$

with weights $w_l$ corresponding to precision expectations. Attention emerges as the adaptive modulation of $w_l$, allocating computational resources to levels where stability change is maximal (Dayan & Abbott 2001).

6. Discussion

The equivalence of stability maximization and free-energy minimization yields several implications.

1. Unified principle of adaptation. Learning, perception, and action can be viewed as the pursuit of informational stability across hierarchical manifolds.

2. Affective interpretation. Affective states correspond to the local temporal curvature of stability; pleasure and displeasure mark acceleration or deceleration toward equilibrium.

3. Robustness and disorder. Cognitive disorders can be interpreted as failures of stability control—either excessive rigidity (over-stabilization) or volatility (under-stabilization)—a view consistent with neurocomputational accounts of schizophrenia and anxiety (Hohwy 2013; Friston 2017).

7. Conclusion

Information-theoretic stability provides a geometric formalism for understanding cognition as an inference process that minimizes divergence between predicted and observed information states. When expressed in predictive-processing terms, the stability reward is identical to the negative derivative of variational free energy, furnishing a compact and rigorous bridge between statistical mechanics, information geometry, and neurocognitive dynamics. This synthesis suggests that cognitive behavior is fundamentally the enactment of stability maintenance within an informational manifold.

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