Continuous basic model

The continuous basic model is a model that describes neural networks . It is much easier than z. B. the Hodgkin-Huxley model , therefore artificial neural networks are often modeled by this model or a discretized version, the discrete basic model.

In the continuous basic model, the individual ion channels of the synapses are no longer modeled, and therefore no individual action potential spikes are visible anymore; instead, these must be explicitly specified by a function .

Thus, every neuron is described by two model equations ( differential equations ):

1. a DGL for the description of the dendritic membrane potential ${\displaystyle x_{j}(t)}$
2. a function evaluation for the axonal potential ${\displaystyle y_{j}(t)=f(x_{j}(t))}$

In a neural network with n neurons, the model equations are then: Where:
${\displaystyle \tau {\dot {x}}_{j}(t)=-x_{j}(t)+u_{j}(t)+\sum _{i=1}^{n}{c_{ij}y_{i}(t-\Delta _{ij}})}$
${\displaystyle y_{j}(t)=f_{j}(x_{j}(t))}$

• ${\displaystyle \tau >0}$ a time constant
• ${\displaystyle x_{j}(t)}$the dendritic potential of the jth neuron
• ${\displaystyle {\dot {x}}_{j}(t)}$ the (temporal) derivation of ${\displaystyle x_{j}}$
• ${\displaystyle u_{j}(t)}$ the external input of the jth neuron
• ${\displaystyle c_{ij}}$ the synaptic coupling strength of the i-th to the j-th neuron
• ${\displaystyle \Delta _{ij}}$ the running time of an action potential from i to j
• ${\displaystyle y_{j}(t)}$the axonal potential of j
• ${\displaystyle f_{j}}$ a transfer function