Figure 1. The passive electrical elements of the cell membrane can be modelled as a simple electric circuit. (a) The cell membrane and embedded ion channels form the membrane capacitance (C_m) and resistance (R_m), respectively. The + and – signs represent charges separated across the capacitor. The double-headed arrows indicate the movement of ions across the membrane through ion channels. (b) The cell membrane can be modelled as an equivalent electrical circuit with R_m and C_m arranged in parallel. The interlocking yellow circles represent a current source that generates I_applied (solid black arrow). This current is split into ionic (I_i, orange arrow) and capacitative (I_c, blue arrow) components by the parallel arrangement of the circuit. 1 shows the current flow during I_applied and 2 shows current flow after I_applied is shut off (dashed line). The current flow after terminating current application (2) results from charge stored in C_m flowing out through R_m. (c) Membrane capacitance delays the change in membrane voltage in response to current injection. The traces show the membrane voltage (V_m, solid black line) response to I_applied (solid black line). The dashed V_m line shows the change in membrane voltage if the capacitor were removed from the circuit. The component currents flowing to the capacitor (I_c) and through the resistor (I_i) are shown as blue and orange traces, respectively.

Figure 2. Neurophysiological impact of membrane . The postsynaptic activity of two neurons with different (top panel) is shown in response to the same train of presynaptic action potentials (bottom panel). The PSP of the neuron, with the 1 ms larger and shorter (blue) than that of the neuron with the 10 ms red, even though the synaptic current (I_applied) is identical in both neurons. The dashed orange line indicates the time course of a single postsynaptic potential in the neuron with =10 ms. Adapted from Kandel E R and Schwartz J H (1981) Principles of Neural Sciences, p. 47. New York: Elsevier/North Holland.

Figure 3. Equivalent membrane electrical circuit showing current flow along the membrane with distance. Recall that only current flowing through the membrane resistance contributes to V_m. In addition, any current crossing the membrane will be lost to subsequent membrane segments. For this reason, V_m decreases with distance from the current source. I_c is not shown because we are assuming that the membrane potential is at steady state (dV/dt=0). R_a and I_a are the axial resistance and axial current, respectively.

Figure 4. Passively propagated electrical signals decay with distance. (a) An axon or dendrite can be modelled as a cable (cylinder) with a conducting core (cytoplasm) surrounded by a leaky insulator (membrane with ion channels) placed into a conducting medium (extracellular solution). A current passing electrode (blue) provides brief current pulses, while a recording electrode (brown) is moved progressively farther from the current source to record the decrease in V_m with distance. The arrows indicate the decrease in current density with distance. (b) The V_m recorded at each electrode position in response to the same stimulus current. The amplitude and rate of rise of V_m decreases with distance, but the duration of the voltage response increases with distance. (c) The equivalent electrical circuit for the cable model. As in Figure 3, the neuron is modelled as a series of parallel r_m c_m segments separated by a resistor (r_a) representing the cytoplasmic resistance. The extracellular resistance is ignored in this example. (d) A plot of the normalized V_m with distance from the current source. This exponential relationship is represented by eqn (7). The determination of is illustrated as the distance where V_m decreases to 37% of its original value.

Figure 5. Differences in can have a strong impact on neuronal signalling. (1) An action potential in the presynaptic neuron (A) simultaneously induces the synaptic release of excitatory neurotransmitter on to two postsynaptic neurons. (2) The PSPs are identical at the postsynaptic site on each neuron (B and C). For the two postsynaptic neurons =10 and 1 mm for B and C, respectively. (3) The PSPs have very different amplitudes after travelling 1 mm to axon hillock (hatched area). The PSP in neuron C has decreased to 37% of its initial amplitude since the distance equals . However, the PSP in neuron B is only slightly decreased since the distance travelled is only 0.1. (4) The PSP in neuron B has sufficient amplitude at the axon hillock to trigger an action potential (AP), while the PSP in neuron C is subthreshold for AP generation. Adapted from Kandel E R and Schwartz J H (1981) Principles of Neural Sciences, p. 49. New York: Elsevier/North-Holland.