Synapses, Current Models

Delta

Formulation

\[ I(t) = Q \delta(t - t_f) \]

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

\[\begin{split} \delta(x) = \begin{cases} 1 / \Delta t & x = 0 \\ 0 & x \neq 0 \end{cases} \end{split}\]

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

  1. ISBN:9780262548083

Single Exponential

Formulation

\[ \tau \frac{dI(t)}{dt} = -I(t) + Q \sum_{\mathcal{F}} \delta \left(t - t_f\right) \]

With solution:

\[ I(t + \Delta t) = I(t) \exp\left(-\frac{\Delta t}{\tau}\right) + \frac{Q}{\tau}\left[t = t_f\right] \]

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

  1. DOI:10.3390/brainsci12070863

Double Exponential

Formulation

\[\begin{split} \begin{align*} \frac{dI(t)}{dt} &= -\frac{I(t)}{\tau_d} + A(t) \\ \frac{dA(t)}{dt} &= -\frac{A(t)}{\tau_r} + \frac{Q}{\tau_r \tau_d} \sum_{t_f < t} \delta \left(t - t_f\right) \end{align*} \end{split}\]

Alternatively:

\[ \tau_d\tau_r \frac{d^2 I(t)}{dt^2} + (\tau_d + \tau_r) \frac{dI(t)}{dt} = -I(t) + Q \sum_{t_f < t} \delta \left(t - t_f\right) \]

With solutions:

\[\begin{split} \begin{align*} I(t + \Delta t) &= \left[I(t) + \frac{\tau_d\tau_r}{\tau_d - \tau_r} A(t) \right] \exp\left(-\frac{\Delta t}{\tau_d}\right) - \frac{\tau_d\tau_r}{\tau_d - \tau_r} A(t + \Delta t) + \frac{Q}{\tau_d - \tau_r}\left[t = t_f\right] \\ A(t + \Delta t) &= A(t) \exp\left(-\frac{\Delta t}{\tau_r}\right) + \frac{Q}{\tau_d\tau_r}\left[t = t_f\right] \end{align*} \end{split}\]

Equivalently:

\[\begin{split} \begin{align*} I(t) &= I_d(t) - I_r(t) \\ I_d(t + \Delta t) &= I_d(t) \exp \left(-\frac{\Delta t}{\tau_d}\right) + \frac{Q}{\tau_d - \tau_r} \left[t = t_f\right] \\ I_r(t + \Delta t) &= I_r(t) \exp \left(-\frac{\Delta t}{\tau_r}\right) + \frac{Q}{\tau_d - \tau_r} \left[t = t_f\right] \end{align*} \end{split}\]

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.

\[t_\text{peak} = \frac{\tau_d\tau_r}{\tau_d - \tau_r} \ln\left(\frac{\tau_d}{\tau_r}\right)\]

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

\[K = \frac{\tau_d - \tau_r}{\exp(-\frac{t_\text{peak}}{\tau_d}) - \exp(-\frac{t_\text{peak}}{\tau_r})}\]

References

  1. DOI:10.3390/brainsci12070863

  2. DOI:10.3389/fncom.2022.806086