Linear Electrical Circuit Theory

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We can think of a length of dendrite or axon to be a stretch of linear cable, visualized as a cylinder made up of cell membrane as the surface of the cylinder and axoplasm (or dendroplasm/cytosol) as the internal medium. If this process is removed and placed in a bath of Ringer or oil, we can think of this cylinder with a small external resistance compared to the resistance of the cell membrane. The figure below illustrates this one-dimensional cable model and its electrical equivalent circuit.I m = C m · + V / R m . (3) When a sudden step of current is applied across this circuit, V = I m · R m {1-e (-t/τ m ) }, (4) where τ m = R m C m and τ m is referred to as the membrane time constant. Linear Cable Theory We can think of a length of dendrite or axon to be a stretch of linear cable, visualized as a cylinder made up of cell membrane as the surface of the cylinder and axoplasm (or dendroplasm/cytosol) as the internal medium. If this process is removed and placed in a bath of Ringer or oil, we can think of this cylinder with a small external resistance compared to the resistance of the cell membrane. The figure below illustrates this one-dimensional cable model and its electrical equivalent circuit. If a current is applied across the cell membrane and if no current leaks through the membrane, then the relationship between the axial current I a and the intracellular voltage V i is given by Ohm’s law, Δ V i / Δx = – r i Α i a , (5) where x is the distance along the cable and r a is the intracellular resistance per unit length of cable. This equation applies only when the membrane resistance is infinite and no current leaks through the membrane. Of course, in reality, membranes have a finite resistance and this parameter has a lot to say about how membrane current is distributed along any cell process and cannot be ignored. This implies that i a will not be constant over any finite length. If we then consider an infinitesimal distance ∂x, across which a voltage difference ∂V occurs.

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