Cellular Neurobiology Equations

9.09J/7.29J - Cellular Neurobiology, Spring 2005 Massachusetts Institute of Technology Department of Brain and Cognitive Sciences Department of Biology Instructors: Professors William Quinn and Troy Littleton 7.29 / j9.09 Cellular Neurobiology List of Equations (plus helpful facts): 1) Ohms law V = IR I = gV g = conductance = l/R; 1 Siemen (5)=1 ohm-l 2) Definition of capacitance Q = CV C = capacitance -a defined constant Q = charge 3) Differentiated definition of capacitance I = dQ/ dt = CdV / dt 4} The Nernst equation: Shown here for potassium Vm = EK = RT /zF .in [K+]0/[I<+]i V m = voltage across membrane EK = Nernst equilibriwn potential for potassium ions R = gas law constant T = temp in °K z = charge number z = I for K+; z = 2 for Ca++ F = Faraday constant = charge (coulombs) on 1 mole of protons For z ~ 1; T -25°C Vm = 58 mV .loglO [K+]0/[K+]i 5} The Goldman equation (for resting potential} Vm = 58 mV .loglO [K+]0 + PNa/PK( [Na+]o + PC1/PK[C1-]i [K+]i + PNa/PK [Na+]i + PC1/PK [C1-]o PNa/PK = Permeability of the cell membrane to sodium ions relative to its permeability to potassium Ions 1 6) Ohm’s law for membranes Im = gk (Vm − EK ) + gNa (Vm − ENa ) current through membrane inward current is defined as negative by the conventions of the textbook. gk = membrane conductance to potassium ions gNa = membrane conductance to sodium ions ENa = Nernst equilibrium potential for sodium ions Im = 7) The Weighted-average equation Vm = gk EK + gNaENa gk + gNa This equation is derived from equation 6) above for the equilibrium condition Im = 0. It describes the same situation as the Goldman Equation; it is less accuracy but easier to use experimentally. Hodgkin & Huxley use it all of the time. 8) The Hodgkin-Huxley predictive cycle Voltage-gated Channels gk, gNa Equation 6 Vm Equation 3 Ik, INa Figure by MIT OCW. 2 9) Passive spread of current in leaky cable – decrease in voltage excursion with distance V(x) = V(o)e− x / λ x = distance from current source λ = space const. = distance for voltage to drop to i/e = 37% of its value at the source 10) λ= rm ri + ro rm 11) = membrane resistance per unit length (Eg cm) of axon For a cylindrical (with arbitrary shaped cross-section) solid, the resistance to Current flow through the cylinder I A = R.l/A l = resistance = specific resistivity a property of the material l = length of solid = cross sectional area of the solid Figure by MIT OCW. * Note that in chapter 6 only resistance = r (lower case) and resistivity = R (upper case). In other chapter R = resistance. Also, charge Q and current I become q and i in chapter 6 only. I don’t understand this change in notation, but students get confused if my lectures depart from it. 12) Definitions for Quantal analysis v1 = mean quantal size (recorded postsynaptically, measured in millivolts) m = mean quantal content (average number of quanta per synaptic stimulation – Measured in quanta) n = number of quanta (vesicles?) available for release at a synapse p = probability of a given individual quantum being released at a given stimulation When n = small – the binomial distribution applies: 3 13) n-x P(x) = n!/x!(n-x)! pX (l-p) so when n = small the probability of failures P(o) = (l-p)x When n = large, we use the Poisson distribution which you need not memorize. From this, the probability of failures (n = large) 14) -m P(o) = e Some facts. quantal analysis distinguishes bet\4/een presynaptic and postsynaptic effects. A. Presynaptic change -> change in m. Most easily measured by measuring change in Po, the rate of failures in stimulated evoked synaptic transmission. B. Postsynaptic change --> change in V1 = change in quanta! size -most easily measured as change in peak voltage for spontaneous mini's. 4