# Synapses, Current Models ## Delta ### Formulation $$ I(t) = Q \delta(t - t_f) $$ *Where the Dirac delta function $\delta(x)$ in discrete time simulations is:* $$ \delta(x) = \begin{cases} 1 / \Delta t & x = 0 \\ 0 & x \neq 0 \end{cases} $$ *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](https://mitpress.ublish.com/ebook/modeling-neural-circuits-made-simple-with-python-preview/12788/Cover) ## 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](https://www.mdpi.com/2076-3425/12/7/863) ## Double Exponential ### Formulation $$ \begin{align*} \frac{dI(t)}{dt} &= -\frac{I(t)}{\tau_d} + A(t) \\ \frac{dA(t)}{dt} &= -\frac{A(t)}{\tau_r} + \frac{Q}{\tau_r \tau_d} \sum_{t_f < t} \delta \left(t - t_f\right) \end{align*} $$ *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:* $$ \begin{align*} I(t + \Delta t) &= \left[I(t) + \frac{\tau_d\tau_r}{\tau_d - \tau_r} A(t) \right] \exp\left(-\frac{\Delta t}{\tau_d}\right) - \frac{\tau_d\tau_r}{\tau_d - \tau_r} A(t + \Delta t) + \frac{Q}{\tau_d - \tau_r}\left[t = t_f\right] \\ A(t + \Delta t) &= A(t) \exp\left(-\frac{\Delta t}{\tau_r}\right) + \frac{Q}{\tau_d\tau_r}\left[t = t_f\right] \end{align*} $$ *Equivalently:* $$ \begin{align*} I(t) &= I_d(t) - I_r(t) \\ I_d(t + \Delta t) &= I_d(t) \exp \left(-\frac{\Delta t}{\tau_d}\right) + \frac{Q}{\tau_d - \tau_r} \left[t = t_f\right] \\ I_r(t + \Delta t) &= I_r(t) \exp \left(-\frac{\Delta t}{\tau_r}\right) + \frac{Q}{\tau_d - \tau_r} \left[t = t_f\right] \end{align*} $$ *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](https://www.mdpi.com/2076-3425/12/7/863) 1. [DOI:10.3389/fncom.2022.806086](https://www.frontiersin.org/journals/computational-neuroscience/articles/10.3389/fncom.2022.806086/full)