# Synapses, Current Models¶

## Delta¶

### Formulation¶

*Where the Dirac delta function \(\delta(x)\) in discrete time simulations is:*

*Where:*

\(I\), total input current applied to the neuron \((\text{nA})\)

\(Q\), electrical charge carried by an action potential \((\text{pC})\)

\(t\), current time of the simulation \((\text{ms})\)

\(t_f\), time of the last presynaptic spike \((\text{ms})\)

\(\Delta t\), length of time over which each simulation step occurs \((\text{ms})\)

Note that the Dirac delta function for a value \(x\) has units inverse of \(x\). So in this case, \(\delta(t - t_f)\) has units \(\text{ms}^{-1}\).

### Description¶

This is a very simplified model for a synapse. In simulations its role is to normalize the current delivered for a given spike such that simulation time step does not dramatically change the integration of current from action potentials into the neuron.

### References¶

## Single Exponential¶

### Formulation¶

*With solution:*

*Where:*

\(I\), total input current applied to the neuron \((\text{nA})\)

\(Q\), electrical charge induced by an action potential \((\text{pC})\)

\(\tau\), time constant of current decay \((\text{ms})\)

\(t\), current time of the simulation \((\text{ms})\)

\(t_f\), time of the last presynaptic spike \((\text{ms})\)

\(\Delta t\), length of time over which each simulation step occurs \((\text{ms})\)

\(\delta(t)\), Dirac delta function \((\text{ms}^{-1})\)

\([\cdots]\) is the Iverson bracket and equals \(1\) if the inner statement is true and \(0\) if it is false (unitless).

### Description¶

This model of synaptic kinetics assumes that the neurotransmitters responsible for inducing the synaptic current instantly bind to receptors and that their influence decays exponentially over time.

### References¶

## Double Exponential¶

### Formulation¶

*Alternatively:*

*With solutions:*

*Equivalently:*

*Where:*

\(I\), total input current applied to the neuron \((\text{nA})\)

\(A\), rate of current change on an action potential \((\text{nA} \cdot \text{ms}^{-1})\)

\(Q\), electrical charge induced by an action potential \((\text{pC})\)

\(\tau_d\), slow time constant of current decay \((\text{ms})\)

\(\tau_r\), fast time constant of current rise \((\text{ms})\)

\(t\), current time of the simulation \((\text{ms})\)

\(t_f\), time of the last presynaptic spike \((\text{ms})\)

\(\Delta t\), length of time over which each simulation step occurs \((\text{ms})\)

\(\delta(t)\), Dirac delta function \((\text{ms}^{-1})\)

Under the condition \(\tau_d > \tau_r > 0\).

### Description¶

This models a synapse where the binding of neurotransmitters is not instantaneous, but instead has both a rise and fall, each controlled by a separate exponential term. For instance, this is typically used when modelling synapses with AMPA and NMDA receptors. AMPA activity is mediated by \(\tau_r\) and NMDA is mediated by \(\tau_d\), where \(\tau_d\) is roughly an order of magnitude larger than \(\tau_r\).

The peak current following a spike at time \(t_f\) is equal to \(t_\text{peak} + t_f\) where \(t_\text{peak}\) is computed as follows.

This is also used to determine the value for \(Q\) for which the peak current will equal 1.