.. index:: ! continuous-time
.. index:: recurrent
.. index:: ! ctrnn
The default :term:`continuous-time` :term:`recurrent` neural network (CTRNN) :py:mod:`implementation <ctrnn>` in neat-python is modeled as a system of ordinary differential equations, with neuron potentials as the dependent variables.
\tau_i \frac{d y_i}{dt} = -y_i + f_i\left(\beta_i + \sum\limits_{j \in A_i} w_{ij} y_j\right)
Where:
- \tau_i is the time constant of neuron i.
- y_i is the potential of neuron i.
- f_i is the :term:`activation function` of neuron i.
- \beta_i is the :term:`bias` of neuron i.
- A_i is the set of indices of neurons that provide input to neuron i.
- w_{ij} is the :term:`weight` of the :term:`connection` from neuron j to neuron i.
The time evolution of the network is computed using the forward Euler method:
y_i(t+\Delta t) = y_i(t) + \Delta t \frac{d y_i}{dt}