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Vm = Vin – Vout V = IR V = I/G Ix = Gx

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Vm = Vin – Vout V = IR V = I/G Ix = Gx Bi 360 - NEUROBIOLOGY EQUATIONS NEUROBIOLOGY EQUATIONS 1. Membrane potential (Vm) Vm = Vin – Vout where Vm is the membrane potential, Vin is the potential on the inside of the cell, and Vout is the potential on the outside of the cell. 2. Ohm’s Law (and derivations) V = IR where V is voltage (measured in volts), I is current (measured in amperes or amps), and R is resistance (measured in ohms). Because conductance is the inverse of resistance (R), Ohm’s law can be rewritten as: V = I/g where g is conductance (measured in mhos or Siemens). The most frequently used derivation of Ohm’s Law in neurobiology rearranges the terms to measure current with respect to conductance and voltage: Ix = gx (Vm – Ex) Where Ix is the current flow for ion X, gx is the conductance for ion X, Vm is membrane potential, Ex is the Nernst or Equilibrium Potential for ion X, and Vm – Ex is the driving force for ion X. 3. Nernst Equation (also known as the Nernst Potential or the Equilibrium Potential) { } Ex = ln { } 푅푇 푋 표푢푡 푧퐹 푋 푖푛 where Ex is the equilibrium potential for ion X, R is the gas constant, T is temperature (degrees Kelvin), z is the valence of ion X, and [X]out and [X]in are the concentrations of ion X outside and inside the cell, respectively. 1 Bi 360 - NEUROBIOLOGY EQUATIONS Since RT/F is 25 mV at 25oC (room temperature), and the constant for converting from natural logarithm to base 10 logarithm is 2.3, the Nernst equation can be written as: { } Ex = { } 58 푚푉 푋 표푢푡 푖푛 푧 푙표푔 푋 4. Goldman Equation (for the resting membrane potential) { } { } { } Vm = { } { } { } 58 푚푉 PK 퐾 표푢푡 + PNa 푁푎 표푢푡 + PCl 퐶푙 푖푛 K 푖푛 Na 푖푛 Cl 표푢푡 푧 푙표푔 P 퐾 + P+ 푁푎+ + P- 퐶푙 where PK, PNa and PCl = permeabilities for K , Na and Cl ions, respectively. 5. Capacitance Q = CV where Q is charge in coulombs, C is capacitance in farads, and V is voltage in volts. Because current (I) is the flow of charge per unit time, then current flow across a capacitor can be determined by the derivative of the above equation: = = 푑푄 푑푉 퐼 퐶 6. Time constant (tau ; the time required푑푡 for푑 the푡 membrane potential to rise to 63% of its steady state value) 휏 The time constant is obtained from the equation that describes the rising phase of the change in membrane potential over time: = (1 ) 푑푉푚 −푡⁄휏 where is the rate of change in 퐼membrane푚 푅푚 − potential푒 (Vm) over time, is 푑푉푚 푑푡 membrane current, is membrane resistance, t is time, and is the time 푑푡 퐼푚 constant. is further defined as: 푚 푅 = 휏 휏 푚 푚 where is membrane resistance휏 and 푅 퐶 is membrane capacitance. 푚 푚 푅 2 퐶 Bi 360 - NEUROBIOLOGY EQUATIONS 7. Length constant (lambda, ; the distance along the membrane for membrane potential to decay to 37% (1/e) of its original value) 휆 The length constant is obtained from the equation that describes the decay in the amplitude of the membrane potential over distance to a current injection at point 0: = −푥⁄휆 푚푥 푚0 where is the membrane potential푉 at푉 distance푒 , is the membrane potential at the site of current injection ( = 0), is distance from the site of 푚푥 푚0 current푉 injection, and is the length constant. is푥 further푉 defined as: 푥 푥 휆 = 휆 푚 푖 Where is membrane resistance휆 and� 푅 is⁄ internal푅 or cytoplasmic resistance. 푅푚 푅푖 3 .
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