= Synaptic potential
{wiki}
A <synaptic potential> is a postsynaptic change in <membrane potential> caused by activity at a <synapse>. Unlike the regenerative <action potential>, a passive <synaptic potential> is graded: its size depends on input strength, the initial <membrane potential> and the receiving <cell membrane>'s properties. Local voltage changes can decay before reaching the <soma>; their timing and overlap help determine whether an <action potential> follows. See <Molecular Biology of the Cell, 4th edition by Bruce Alberts and others/Ion Channels and the Electrical Properties of Membranes>.
<Depolarization> means that the interior becomes less negative relative to the exterior; <hyperpolarization> means that it becomes more negative. Excitation and inhibition instead describe whether an input promotes or opposes postsynaptic <action potential> generation. An <inhibitory postsynaptic potential> can be depolarizing, and <shunting inhibition> can occur with no voltage change at rest. The <receptor>, permeating <ions>, ionic gradients and <reversal potential> matter. See https://www.ncbi.nlm.nih.gov/books/NBK11117/[Purves and colleagues: excitation, inhibition and depolarizing inhibition].
A worked passive model treats the receiving membrane as one compartment with <capacitance> $C>0$, leak conductance $g_L>0$, and activated synaptic conductances $g_e,g_i\geq0$:
$$
C\frac{dV}{dt}=g_L(E_L-V)+g_e(E_e-V)+g_i(E_i-V)+I_{\mathrm{ext}}.
$$
All <electric currents> are inward-positive. Each term $g(E-V)$ vanishes at its <reversal potential> $E$; $\mathrm{nS}\times\mathrm{mV}=\mathrm{pA}$. With constant parameters, the <ordinary differential equation> has
$$
G=g_L+g_e+g_i,\qquad V_\infty=\frac{g_LE_L+g_eE_e+g_iE_i+I_{\mathrm{ext}}}{G},\qquad \tau=\frac{C}{G},
$$
$$
V(t)=V_\infty+(V(0)-V_\infty)e^{-t/\tau}.
$$
Here $G$ is total conductance, $1/G$ is effective <electrical resistance>, $V_\infty$ is the <steady state> and $\tau$ the <time constant>. Without imposed <electric current>, $V_\infty$ is a conductance-weighted average of the <reversal potentials>. This approximation omits spatial structure, changing ionic gradients and active <voltage-gated ion channels>; it calculates passive voltage responses, not actual <action potential> waveforms. See <Neuronal Dynamics: From Single Neurons to Networks and Models of Cognition by Wulfram Gerstner and others/3.1 Synapses> for the mechanism behind the terms; the inward-positive convention here reverses that source's synaptic-current sign.
Take the same illustrative leak and <capacitance> as under <shunting inhibition>, start at $V(0)=-65\,\mathrm{mV}$ and set $I_{\mathrm{ext}}=0$. Opening $g_e=2\,\mathrm{nS}$ with $E_e=0\,\mathrm{mV}$ gives $V_\infty=-650/12\simeq-54.167\,\mathrm{mV}$ and $\tau\simeq16.667\,\mathrm{ms}$. Its initial inward <electric current> is $2(0+65)=130\,\mathrm{pA}$. Adding $g_i=20\,\mathrm{nS}$ with $E_i=-65\,\mathrm{mV}$ changes $V_\infty$ to $(-650-1300)/32=-60.9375\,\mathrm{mV}$ and $\tau=6.25\,\mathrm{ms}$. This is <shunting inhibition> of the same excitatory drive.
For comparison, $g_i=20\,\mathrm{nS}$ with $E_i=-75\,\mathrm{mV}$ alone gives $V_\infty\simeq-71.667\,\mathrm{mV}$: <hyperpolarization>. With $E_i=-60\,\mathrm{mV}$ instead, it alone gives $V_\infty\simeq-61.667\,\mathrm{mV}$: <depolarization>. Yet adding this depolarizing conductance to the $g_e=2\,\mathrm{nS}$ excitatory drive lowers that drive's $V_\infty$ from $-54.167$ to $(-650-1200)/32=-57.8125\,\mathrm{mV}$. At an illustrative firing threshold $V=-50\,\mathrm{mV}$, its <electric current> is $20(-60+50)=-200\,\mathrm{pA}$, opposing threshold approach. The combined trace initially rises faster than the excitatory drive alone, before settling below it. Thus the sign of a voltage change at rest alone does not determine excitation or inhibition; a claim about firing must specify input strength, timing and the <neuron>'s state.
\Image[/synaptic-conductance-and-shunting.png]
{title=<Reversal potential>, passive <synaptic potentials> and <shunting inhibition>}
{description=Computed illustrations, not recordings. Left: inward <electric current> for a $1\,\mathrm{nS}$ conductance with four <reversal potentials>. Upper right: constant-conductance steps at time zero in the one-compartment model with $C=200\,\mathrm{pF}$, $g_L=10\,\mathrm{nS}$ and $E_L=V(0)=-65\,\mathrm{mV}$. Lower right: the same $100\,\mathrm{pA}$ test <electric current> with and without $20\,\mathrm{nS}$ of added conductance at $E_i=-65\,\mathrm{mV}$. All plotted trajectories are passive; no <action potential> generation or reset is included.}
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