PIATRA . INSTITUTE

trainable gene circuits

associative memory, bistable commitment, and oscillatory state in gene regulatory networks

based on 2022, Michael Levin et al., Learning in Transcriptional Network Models: Computational Discovery of Pathway-Level Memory and Effective Interventions

conditioning-like response

The probe stimulus evokes a strong output because the slow variable retained a training trace.

peak A: 3.00

peak C: 3.00

probe mean A: 2.21

time-series: A (output) / B (gate) / C (memory) / U1, U2 (stimuli)

final A

1.707

final B

1.866

final C

3.000

probe mean A

2.210

training

probe

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:

\dot{x}_i = f_i(x_1, \dots, x_n, u) - \gamma_i x_i

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:

f_{\text{act}}(A) = \beta \frac{A^n}{K^n + A^n} \qquad f_{\text{rep}}(C) = \beta \frac{K^n}{K^n + C^n}

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:

\dot{p} = f_p(w_1, w_2, u_1, u_2) - \delta_p \, p

\dot{w}_2 = f_2(p, u_2) - \delta_w \, w_2

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.

claude opus 4.8March 2026·first cut. three coupled production-degradation ODEs with shared Hill regulation, integrated by fourth-order Runge-Kutta under a five-phase stimulation protocol (baseline, training, washout, probe, after). three presets cover the canonical learning regimes: a slow-variable associative trace, a bistable toggle, and a repressilator oscillator. a narrative panel labels each run, and the calibration verifies the deterministic primitives plus the conditioning, commitment, and oscillation outcomes against recomputed targets.

Calibration

Assumptions

Model changelog

v1.0March 2026

three-gene model: dx_i/dt = basal_i + beta_i * regulation_i(x, u) - decay_i * x_i with signed Hill regulation on every edge.
five-phase stimulation protocol so a transient training pulse can be probed after a washout, separating durable memory from transient response.
associative preset where gene C is a low-decay slow variable that stores paired stimulation and gates the output, after the Fernando-Levin conditioning circuit.
toggle preset using mutual repression plus self-activation for bistable basin commitment, and a repressilator preset that stores state as a sustained oscillation.
narrative inference labels each run as conditioning-like, committed, oscillatory, or no durable learning, computed from probe-window statistics.
calibration: exact Hill half-saturation and zero-weight identities, plus deterministic conditioning gain, toggle dominance, and oscillation amplitude recomputed from the live logic.
framing kept honest: learning means history-dependent state change under timed stimulation, not topology rewiring or gradient descent.

read the research companion →