GRNs as Dynamical Systems

A gene regulatory network is a dynamical system on expression space. Each gene product's concentration $x_i$ evolves according to a nonlinear ODE:

where $f_i$ encodes regulatory inputs (activation and repression via Hill functions) and $\gamma_i x_i$ is degradation. Cell types are modeled as stable attractors of this system.

Hill Function Regulation

Activation and repression use saturating Hill functions:

The Hill coefficient $n$ controls switch-like sharpness. Higher $n$ gives a more digital response. The threshold $K$ sets the inflection point.

Associative Learning in GRNs

Following Levin and Fernando, the associative circuit uses a slow memory variable $C$ that integrates co-stimulation:

The crucial feature: $w_2$ is a slow state variable that stores the effect of prior co-stimulation. After training, a formerly weak cue can drive the response. This is the biochemical analog of a synaptic weight.

Three Memory Regimes

• Associative trace: a slow variable retains history of co-stimulation. Later, a weak cue triggers the trained response. Learning = threshold crossing into a new basin.
• Toggle / bistable attractor: mutual repression with self-activation creates two stable states. Memory = which basin the system occupies after a transient pulse.
• Oscillatory memory: a repressilator-style loop stores state in phase and amplitude rather than a fixed point. Memory = persistent dynamical regime.

Notes

• Simulation uses fourth-order Runge-Kutta (RK4) integration.
• “Learning” here means history-dependent state change, not gradient descent on parameters.
• The key insight: a GRN can exhibit memory, conditioning, and trainable behavior through the dynamics of its existing equations, without rewiring its topology.
• Values are illustrative. Replace coupling weights and timing to test specific biological circuits.