Cellular Neuroscience (207) Ian Parker Lecture # 2 - Passive electrical properties of membranes (What does all this electronics stuff mean for a neuron?)

Membrane structure Fatty acid tail polar head Permeation across the lipid bilayer increases with increasing lipid solubility. Ions and water are almost completely impermeant – protein channels and carriers. Provide pathways for selective movement of ions and molecules across the membrane.

The cell membrane (lipid bilayer) acts as a very good insulator, but has high capacitance. (WHY?) Specific membrane resistance 1 cm Resistance of 1 cm 2 of membrane (R m ) R m of a lipid bilayer >10 6  cm 2 But membrane channels can greatly increase the membrane conductance

Specific membrane capacitance membrane extracellular fluid intracellular fluid The insulating cell membrane (dielectric) separates two good conductors (the fluids outside and inside the cell), thus forming a capacitor. Because the membrane is so thin (ca. 7.5 nm), the membrane acts as a very good capacitor. Specific capacitance (capacitance of 1 cm 2 of membrane : C m ) C m ~ 1  F cm -2 for cell membranes

Input resistance of a cell Inject current (I) Record voltage (V) cell Input resistance R in = V/I R in decreases with increasing size of cell (increasing membrane area) R in increases with increasing specific membrane resistance [If I = 10 nA and V = 5 mV, what is Rm ???]

A neuron as an RC circuit Inject current (I) Record voltage (V) cell RmRm CmCm inside outside I Voltage changes exponentially with time constant  m

 m = R m * C m So  m will be longer if R m is high “ “ “ “ “ and if C m is high We can directly measure R m and  m so we can calculate C m =  m / R m Given that C m ~ 1  F cm 2, we can then calculate the membrane area of the cell

EXERCISE From example trace given in class; Measure time constant Measure change in membrane potential resulting from a given injection of current Calculate input resistance Calculate total capacitance of cell membrane Estimate diameter of the cell

But, neurons are not usually spheres! What about axons and dendrites? For a spherical cell, all regions on the cell membrane are at the same potential as each other (isopotential). This is not the case for long, thin processes such as axons and dendrites. The voltage change induced by a ‘square’ pulse of injected current gets smaller and more rounded with increasing distance along an axon

Space constant Space constant ( ) is the distance at which the change in membrane potential (  Vm) falls to 1/e of some initial value. (What is the space constant in the example above?)

Equivalent circuit of an axon The math for this gets complicated … So, we will just consider a simple case where the duration of current injection is long enough for the membrane capacitance to fully charge. Then; (Qualitatively, how would the space constant be affected if the current pulse was brief?)

What determines the space constant (  For long current pulses we can ignore membrane capacitance, so the circuit simplifies to; More current will flow along the inside of the axon (i.e. the voltage change will travel further) if; Ri (resistance of the axoplasm ) is lower Rm (‘leakage’ resistance across the membrane) is greater An analogy: a leaky hosepipe More water will flow out of the end of the pipe if its diameter is greater (fire hose vs garden hose), and if the number of leaks is smaller

How does vary with the diameter of an axon? Membrane resistance Rm is proportional to the area of membrane per unit length. So, Rm is proportional to the circumference of the axon (2  r) Longitudinal axoplasmic resistance (Ri) per unit length is proportional to cross sectional area (  r 2 ) As the diameter of an axon increases Rm decreases linearly, whereas Ri decreases as the square root of the diameter. Because = Sqrt(Rm/Ri) increases proportional to the square root of the axon diameter. Thus, a signal will passively propagate a longer distance in a fat axon than in a skinny axon.