Recurrent neural chemical reaction networks trained to switch dynamical behaviours through learned bifurcations

TL;DR

This work shows that recurrent neural chemical reaction networks (RNCRNs), a class of chemical reaction networks based on recurrent artificial neural networks that can be trained to reproduce a given dynamical behaviour, can be trained to exhibit bifurcations, and introduces an ODE-free algorithm for training the RNCRN to display designer oscillations.

Abstract

Both natural and synthetic chemical systems not only exhibit a range of non-trivial dynamics, but also transition between qualitatively different dynamical behaviours as environmental parameters change. Such transitions are called bifurcations. Here, we show that recurrent neural chemical reaction networks (RNCRNs), a class of chemical reaction networks based on recurrent artificial neural networks that can be trained to reproduce a given dynamical behaviour, can also be trained to exhibit bifurcations. First, we show that RNCRNs can inherit some bifurcations defined by smooth ordinary differential equations (ODEs). Second, we demonstrate that the RNCRN can be trained to infer bifurcations that allow it to approximate different target behaviours within different regions of parameter space, without explicitly providing the bifurcation itself in the training. These behaviours can be specified using target ODEs that are discontinuous with respect to the parameters, or even simply by specifying certain desired dynamical features in certain regions of the parameter space. To achieve the latter, we introduce an ODE-free algorithm for training the RNCRN to display designer oscillations, such as a heart-shaped limit cycle or two coexisting limit cycles.

Figures (12)

• Figure 1: A graphical visualisation of the recurrent neural chemical reaction network (RNCRN) underlying the reaction-rate equations (RREs) (\ref{['eq:single_layer_RRE']}), as introduced in dack_recurrent_2025. Orange circles represent chemical perceptron species $Y_1, \dots, Y_M$, while magenta squares represent the executive species $X_1, \dots, X_N$. Coloured arrows represent chemical reactions consistent with the RREs: blue arrows represent $X_i + Y_j \xrightarrow{|\alpha_{i,j}|} 2X_i + Y_j$ or $X_i + Y_j \xrightarrow{|\alpha_{i,j}| }Y_j$ (according to the sign of $\alpha_{i,j}$), red arrows represent $X_i + Y_j \xrightarrow{|\omega_{j,i}|/\mu} X_i + 2Y_j$ or $X_i + Y_j \xrightarrow{|\omega_{j,i}|/ \mu} X_i$ (according to the sign of $\omega_{j,i}$), magenta arrows represent $\varnothing \xrightarrow{\beta_i} X_i$, and orange arrows represent three reactions $\varnothing \xrightarrow{\gamma/ \mu} Y_j$, $2Y_j \xrightarrow{1/ \mu} Y_j$, and one of $Y_j \xrightarrow{|\theta_j|/ \mu} 2Y_j$ or $Y_j \xrightarrow{|\theta_j|/ \mu} \varnothing$ (according to the sign of $\theta_j$).
• Figure 2: Approximation of two bifurcation-exhibiting ODEs using RNCRNs. Row (a) shows the target ODE (\ref{['eq:target_hopf']}) undergoing a Hopf bifurcation as the parameter $\bar{r}_1$ is varied through its critical value. Row (b) shows the $M=5$ chemical perceptron RNCRN approximation from Appendix \ref{['sec:app_hopf_bif']} with $\mu=0.01$ undergoing the Hopf bifurcation as its equivalent control species $\lambda_1$ is varied. Row (c) shows the target ODE (\ref{['eq:homoclinic_orbit_RRE']}) undergoing a homoclinic bifurcation as the parameter $\bar{r}_1$ is varied. Row (d) shows the $M=10$ chemical perceptron RNCRN approximation from Appendix \ref{['sec:app_homoclinic_bif']} with $\mu=0.0075$ undergoing a homoclinic bifurcation. The gray arrows show the vector field of the target ODEs for rows (a) and (b) while for rows (b) and (d) the arrows show the vector field of the reduced RNCRN (i.e. $\mu=0)$.
• Figure 3: Emergence of a bifurcation via interpolation when an RNCRN is trained at discrete concentrations of a parameter species. Approximation of (\ref{['eq:point_con_piecewise']}) by the parametrized RNCRN (\ref{['eq:toogle_piecewise_ex_rncrn']}) specified in Appendix \ref{['sec:simple_uni_bi_piecewise']}. Panel (a) shows the dynamics in the first dynamical regime ($\lambda_1 = 1$). The target dynamics $d \bar{x}_1 / d t = 30 - 6 \bar{x}_1$ are shown in blue while the reduced vector field of the RNCRN approximation, i.e. when $\mu=0$, are shown in magenta. Panel (b) shows the dynamics in the second dynamical regime ($\lambda_1 = 0$). In blue the target dynamics $d \bar{x}_1 / d t = -\left(\bar{x}_1 - 2\right)(\bar{x}_1 - 5)(\bar{x}_1 - 8)$ while in magenta the reduced vector field of the RNCRN approximation. Panel (c)--(d) show $x_1(t)$ time-trajectories over a variety of executive species initial conditions, $x_1(0) = 2, 3, 6, 7$, for the full RNCRN in each dynamical regime with $\mu=0.1$ and all chemical perceptron initial conditions at zero $y_1(0)=y_2(0)=y_3(0)=0$. Panel (e) shows the bifurcation diagram for the RNCRN approximation as the parameter $\lambda_1$ is varied through intermediate values in magenta. A bifurcation is observed at around $\lambda_1 =10^{-2}$.
• Figure 4: A schematic of the classification-controlled RNCRN used to approximate equation (\ref{['eq:class_con_piecewise']}). In purple the parameter-species $\Lambda_1$ and $\Lambda_2$ are shown to interact with the sense-chemical-perceptron species that perform the non-linear classification presenting the output as the magnitude of the bifurcation species $R$. The bifurcation species $R$ then acts as a parameter in the response-chemical-perceptron species that are trained to induce the intended dynamics in the executive species.
• Figure 5: A panel of $x_1(t)$ trajectories for different concentrations of $\Lambda_1$ and $\Lambda_2$ showing the non-linear XOR classification boundary separating bistability and unistability for an RNCRN approximating the target piecewise system in equation (\ref{['eq:class_con_piecewise']}). The associated classification-controlled RNCRN, i.e. equation (\ref{['eq:modular_sac_RRE']}), uses four sense chemical-perceptrons, one bifurcation chemical-perceptron, and three response chemical-perceptrons to induce the intended dynamics in one executive species according to the states of two parameter species $\Lambda_1$ and $\Lambda_2$. Each panel has two time-trajectories simulated for a total time of $\Delta t=1$ with $\mu=0.001$, all chemical perceptron values starting at zero and the only executive species taking alternative values of $x_1(0) = 2$ and $x_1(0)=7$. The hue of each panel shows the indented dynamical behaviour in the piecewise system: yellow should be unistable while pink should be bistable. Full details of the reactions in this example are given in Appendix \ref{['app:sar_piecewise_struct']}.