Continuous Time Recurrent Neural Networks

CTRNNs are a type of Recurrent Neural Network designed to model temporal dynamics in continuous time. They are particularly useful in fields like evolutionary robotics, where agents need to process sensory inputs and generate motor outputs in real-time.

The most basic definition is as follows:

$${\tau }_i\frac{d{y}_i(t)}{dt}=-y_i(t)+\sum _{j=1}^N{w}_{ij}\sigma (y_j(t)+{\theta }_j)+I_i(t)$$

Where:

• ${\tau }_i$ is the time constant of neuron $i$, determining how quickly it responds to inputs.
• $\frac{d{y}_i(t)}{dt}$ is the rate of change of the neuron’s state over time.
• $-y_i(t)$ represents the decay of the neuron’s state.
• ${\sum }_{j=1}^N{w}_{ij}\sigma (y_j(t)+{\theta }_j)$ is the weighted sum of inputs from other neurons, where $w_{ij}$ is the weight from neuron $j$ to neuron $i$, $\sigma$ is an activation function (often a sigmoid), and ${\theta }_j$ is a bias term.
• $I_i(t)$ is the external input to neuron $i$ at time $t$.

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Difference Equation

For simulation, we update neuron states at each timestep:

$$y_i(t+1)=y_i(t)+\frac{1}{{\tau }_i}\left(-y_i(t)+\sum _{j=1}^N{w}_{ij}\sigma (y_j(t)+{\theta }_j)+I_i(t)\right)$$

The $-y_i(t)$ term: Decay that pulls the state back to zero. This mirrors the biological tendancy towards equilibrium.

Parameter effects:

• ${\tau }_i$ large → slow, integrative
• ${\tau }_i$ small → fast, reactive
• $w_{ij}$ large → strong recurrence
• ${\theta }_j$ positive → neuron more easily activated

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Comparison to BioPhysics Models

CTRNNs abstract away from detailed biophysical models like the Hodgkin Huxley Model, which models ion channel conductances with four coupled ODEs. CTRNNs collapse this complexity into a single state variable with decay, capturing rate dynamics rather than spiking.

I thought it would be of interesting note, because they are similar in quite a few ways. Mean Field Theory was introduced to solve the exact $w_{ij}$ calculation for Hodgkin Huxley as that was a complicated operation.