# Evolution of Activity-Dependent Adaptive Boolean Networks towards Criticality: An Analytic Approach

## TL;DR

This paper introduces activity-dependent adaptive Boolean networks inspired by gene regulation, providing an analytical framework to understand their evolution towards criticality and matching real-world gene network properties.

## Contribution

It presents a novel analytical approach to model adaptive Boolean networks, demonstrating how they can evolve towards criticality with tunable sensitivity, supported by numerical validation.

## Key findings

- Analytical solution for stationary in-degree distribution.
- Network rewiring drives networks towards criticality when sensitivity is set to 1.
- Good agreement between theory and numerical simulations.

## Abstract

We propose new activity-dependent adaptive Boolean networks inspired by the cis-regulatory mechanism in gene regulatory networks. We analytically show that our model can be solved for stationary in-degree distribution for a wide class of update rules by employing the annealed approximation of Boolean network dynamics and that evolved Boolean networks have a preassigned average sensitivity that can be set independently of update rules if certain conditions are satisfied. In particular, when it is set to 1, our theory predicts that the proposed network rewiring algorithm drives Boolean networks towards criticality. We verify that these analytic results agree well with numerical simulations for four representative update rules. We also discuss the relationship between sensitivity of update rules and stationary in-degree distributions and compare it with that in real-world gene regulatory networks.