# iaf_psc_delta – Leaky integrate-and-fire model with delta-shaped input currents¶

## Description¶

iaf_psc_delta is a leaky integrate-and-fire neuron model with

• a hard threshold,

• a fixed refractory period,

• Dirac delta ($$\delta$$)-shaped synaptic input currents.

### Membrane potential evolution, spike emission, and refractoriness¶

The membrane potential evolves according to

$\frac{dV_\text{m}}{dt} = -\frac{V_{\text{m}} - E_\text{L}}{\tau_{\text{m}}} + \dot{\Delta}_{\text{syn}} + \frac{I_{\text{syn}} + I_\text{e}}{C_{\text{m}}}$

where the derivative of change in voltage due to synaptic input $$\dot{\Delta}_{\text{syn}}(t)$$ is discussed below and $$I_\text{e}$$ is a constant input current set as a model parameter.

A spike is emitted at time step $$t^*=t_{k+1}$$ if

$V_\text{m}(t_k) < V_{th} \quad\text{and}\quad V_\text{m}(t_{k+1})\geq V_\text{th} \;.$

Subsequently,

$V_\text{m}(t) = V_{\text{reset}} \quad\text{for}\quad t^* \leq t < t^* + t_{\text{ref}} \;,$

that is, the membrane potential is clamped to $$V_{\text{reset}}$$ during the refractory period.

### Synaptic input¶

The change in membrane potential due to synaptic inputs can be formulated as:

$\dot{\Delta}_{\text{syn}}(t) = \sum_{j} w_j \sum_k \delta(t-t_j^k-d_j) \;,$

where $$j$$ indexes either excitatory ($$w_j > 0$$) or inhibitory ($$w_j < 0$$) presynaptic neurons, $$k$$ indexes the spike times of neuron $$j$$, $$d_j$$ is the delay from neuron $$j$$, and $$\delta$$ is the Dirac delta distribution. This implies that the jump in voltage upon a single synaptic input spike is

$\Delta_{\text{syn}} = w \;,$

where $$w$$ is the corresponding synaptic weight in mV.

The change in voltage caused by the synaptic input can be interpreted as being caused by individual post-synaptic currents (PSCs) given by

$i_{\text{syn}}(t) = C_{\text{m}} \cdot w \cdot \delta(t).$

As a consequence, the total charge $$q$$ transferred by a single PSC is

$q = \int_0^{\infty} i_{\text{syn, X}}(t) dt = C_{\text{m}} \cdot w \;.$

By default, $$V_\text{m}$$ is not bounded from below. To limit hyperpolarization to biophysically plausible values, set parameter $$V_{\text{min}}$$ as lower bound of $$V_\text{m}$$.

Note

Spikes arriving while the neuron is refractory, are discarded by default. If the property refractory_input is set to True, such spikes are added to the membrane potential at the end of the refractory period, dampened according to the interval between arrival and end of refractoriness.

## Parameters¶

The following parameters can be set in the status dictionary.

Parameter

Default

Math equivalent

Description

E_L

-70 mV

$$E_\text{L}$$

Resting membrane potential

C_m

250 pF

$$C_{\text{m}}$$

Capacitance of the membrane

tau_m

10 ms

$$\tau_{\text{m}}$$

Membrane time constant

t_ref

2 ms

$$t_{\text{ref}}$$

Duration of refractory period

V_th

-55 mV

$$V_{\text{th}}$$

Spike threshold

V_reset

-70 mV

$$V_{\text{reset}}$$

Reset potential of the membrane

I_e

0 pA

$$I_\text{e}$$

Constant input current

V_min

$$-\infty$$ mV

$$V_{\text{min}}$$

Absolute lower value for the membrane potential

refractory_input

False

None

If set to True, spikes arriving during refractory period are integrated afterwards

SpikeEvent