An artificial pacemaker an cardiac cells to threshold and initiate action potent

Question

An artificial pacemaker an cardiac cells to threshold and initiate action potentials when the pacemaker cells in the heart are working properly. Suppose that the pacemaker provides a current applied as a square wave pulse (for the purposes of this you can consider it to be a step input). The cardiac muscle cell membrane can be modeled as a resistor and capacitor in parallel, as shown below. We want to bring the membrane potential up from -90 mV to the threshold value of -55 mV (after which Na^+ currents open and the action potential begins), so we need to apply a voltage of 35 mV. The value of the membrane resistance is 3.3 times 10^3 ohm and membrane capacitance is 1.5 times 10^-6 F. If we want the increase in voltage to occur within 5 ms, what should be the value of the current applied (I_m)?

Explanation / Answer

As per the question C is already at -55mV after applying 35mV( increased from -90mV to -55mV).

Since R and C are in parallel, total current Im is summation of current through R and current through C.

So Im = IR+IC

Step 1: Current through R(IR)= 35mV(applied voltage)/3.3k Ohm

IR= 10.606 uA

Step 2: Current through C (IC)= C dV/dT

dV- Change in voltage -90mV to -55mV(-90mV-(-35mV)=35mV)

dT- time duration for the capacitor voltage change

Capacitor initial charge is -90mV before applying 35mV, its voltage has to change from -90mV to -55mV within 5mS(let us take 4.5mS).

Therefore IC= 1.5u* (35m/4.5m)

  IC= 11.67uA

Step 3: so Im = IR+IC  

Im = 10.606uA+11.67uA

Im = 22.276uA

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