Simplest Model of Nervous System. II. Evolutionary Optimization

Abstract

In this work, we build upon a simple model of a primitive nervous system presented in a prior companion paper. Within this model, we formulate and solve an optimization problem, aiming to mirror the process of evolutionary optimization of the nervous system. The formally derived predictions include the emergence of sharp peaks of neural activity (‘spikes’), an increasing sensory sensitivity to external signals and a dramatic reduction in the cost of the functioning of the nervous system due to evolutionary optimization. Our work implies that we may be able to make general predictions about the behavior and characteristics of the nervous system irrespective of specific molecular mechanisms or evolutionary trajectories. It also underscores the potential utility of evolutionary optimization as a key principle in mathematical modeling of the nervous system and offers examples of analytical derivations possible in this field. Though grounded in a simple model, our findings offer a novel perspective, merging theoretical frameworks from nonequilibrium statistical physics with evolutionary principles. This perspective may guide more comprehensive inquiries into the intricate nature of neural networks.

Making Sense of Neural Networks in the Light of Evolutionary Optimization

“…As demonstrated above, equations ( 23)- (25), relying solely on a weak noise limit (σ 2 → 0), as commonly done, 15,18,20,21,[23][24][25]32,34,35 can sometimes produce misleading outcomes even at local minima of the potential energy, let alone the regular functioning of neural networks. In addition, we have previously provided specific examples of artifacts that stem from neglecting finite σ 2 effects in the toy model, namely the divergence of a component of f at the biologically most plausible values of its arguments, and a loss of an arbitrary contribution to Q (the only independent component of ℚ in that toy model) not restricted by the choice of the other component of f. 27,28 Taken together, these results on the weak noise limit highlight the need for a more rigorous mathematical approach in the transition from the Fokker-Planck representation to the representation of a system in terms of the functions u and ℚ.…”

“…This encompasses the sharp excitation and relaxation responses of neurons, the energetic efficiency of neural system operations after evolutionary optimization, and the connection between global and local optimization strategies (higher population growth rate vs. faster kinetics and greater sensitivity of ion channels), as outlined for the general case in this work, and demonstrated previously for a particular simple case. 27,28 However, challenges remain. The framework, as it currently stands, is not fully comprehensive.…”

“…Necessary and sufficient constraints on u and ℚ might be introduced by demanding that a stationary solution to the Fokker-Planck equation Pstat exists for dynamic equations ( 4) with f defined from u and ℚ by equation (18). The analysis of the toy model 27,28 demonstrates that the mere existence of u with necessary properties like condition ( 43) is not sufficient for the existence of Pstat, despite equation (15). With some values of ℚ (for example, the above-mentioned unacceptable solution for the toy model) the stationary distribution may diverge.…”

Simplest Model of Nervous System. I. Formalism

“…The question of evolutionary optimization in the presented model is addressed in a companion paper. 32 Stationary distribution exists. To verify the theoretical prediction on the existence of the stationary distribution, we ran multiple independent numerical simulations of the model, using different initializations of the random number generator.…”

“…where the first term (quadratic in r) is inherited from the asymptote (23), the second term (logarithmic in r) is included by analogy with the effective potential (49), where it corresponds to the 1/r term in f2 omitted in the long-distance asymptotic analysis [see equation (17)], and the third term (exponential in r) is included based on the numerical analysis (data fit with power law or hyperbolic terms, instead of the exponential term, yielded significantly lower accuracy). This approximation (32) provides high accuracy in a wide range of r values (Fig. 3c), including poorly sampled regions of large r (due to the use of the exact asymptote) and small r (because an exponential function happened to provide an accurate fit for the numerical data).…”