Equivalent Circuit Model of the Neuron Generator Potentials, �Synaptic Potentials and Action Potentials All Can Be Described by the Equivalent Circuit Model of the Membrane The �Nerve (or Muscle) Cell can be Represented by a Collection of Batteries, Resistors and Capacitors PNS, Fig 2-11 Equivalent Circuit of the Membrane and Ions Cannot Diffuse Across the Hydrophobic Barrier of the Lipid Bilayer Passive Electrical Properties • Equivalent Circuit of the Membrane – What Gives Rise to C, R, and V? – Model of the Resting Membrane • Passive Electrical Properties – Time Constant and Length Constant – Effects on Synaptic Integration • � Voltage-Clamp Analysis of the Action Potential The Lipid Bilayer Acts Like a Capacitor Capacitance is Proportional to Membrane Area + - - + + V m = Q/C - - - - + + + + + + + - - - - - - - - + + + + - - - - - - - - - - - - ∆ V m = ∆ Q/C + + - - + - - + + + - - - - - - - - - - - - - - + + + + ∆ Q must change before - - ∆ V m can change - - - - + + + - - + 1
Electrical Signaling in the Nervous System is The Bulk Solution Remains Electroneutral Caused by the Opening or Closing of Ion Channels + - - + - - - - + + + - - + - + - - - + + + - - - - + - - + The Resultant Flow of Charge into the Cell Drives the Membrane Potential Away From its Resting Value PNS, Fig 7-1 Each K + Channel Acts as a Conductor Ion Channel Selectivity and Ionic Concentration (Resistance) Gradient Result in an Electromotive Force PNS, Fig 7-5 PNS, Fig 7-3 An Ion Channel Acts Both as a Conductor and as a Battery All the K + Channels Can be Lumped into One Equivalent Structure RT [K + ] o E K = • ln zF [K + ] i PNS, Fig 7-6 PNS, Fig 7-7 2
An Ionic Battery Contributes to V M in Proportion to the When g K is Very High, g K •E K Predominates Membrane Conductance for That Ion The K + Battery Predominates at Resting Potential The K + Battery Predominates at Resting Potential ≈ ≈ g K g K This Equation is Qualitatively Similar to the Goldman Equation The Goldman Equation V m = RT•ln (P K {K + } o + P Na {Na + } o + P Cl {Cl - } i ) • ln V m = zF (P K {K + } i + P Na {Na + } i + P Cl {Cl - } o ) 3
Ions Leak Across the Membrane at At Resting Potential The Cell is in a Resting Potential Steady-State Out In PNS, Fig 7-10 Passive Properties Affect Synaptic Integration Equivalent Circuit of the Membrane and Passive Electrical Properties • Equivalent Circuit of the Membrane – What Gives Rise to C, R, and V? – Model of the Resting Membrane • Passive Electrical Properties – Time Constant and Length Constant – Effects on Synaptic Integration • � Voltage-Clamp Analysis of the Action Potential Experimental Set-up for Equivalent Circuit for Injecting Current into Cell Injecting Current into a Neuron PNS, Fig 8-2 PNS, Fig 7-2 4
If the Cell Had Only Resistive Properties If the Cell Had Only Resistive Properties ∆ V m = I x R in PNS, Fig 8-2 If the Cell Had Only Capacitive Properties If the Cell Had Only Capacitive Properties ∆ V m = ∆ Q/C PNS, Fig 8-2 The Vm Across C is Always Equal to Because of Membrane Capacitance, Vm Across the R Voltage Always Lags Current Flow Out ∆ V m = IxR in ∆ V m = ∆ Q/C τ = R in x C in τ In PNS, Fig 8-2 PNS, Fig 8-3 5
Length Constant λ = √ r m /r a Spread of Injected Current is Affected by r a and r m ∆ V m = I x r m PNS, Fig 8-5 Receptor Potentials and �Synaptic Potentials Synaptic Integration Convey Signals over Short Distances Action Potentials Convey Signals over Long Distances PNS, Fig 2-11 PNS, Fig 12-13 The Action Potential Equivalent Circuit of the Membrane and Passive Electrical Properties 1) Has a threshold, is all-or-none, and is conducted without decrement 2) Carries information from one end of the neuron to the other in a pulse-code • Equivalent Circuit of the Membrane – What Gives Rise to C, R, and V? – Model of the Resting Membrane • Passive Electrical Properties – Time Constant and Length Constant – Effects on Synaptic Integration • � Voltage-Clamp Analysis of the Action Potential PNS, Fig 2-10 6
Sequential Opening of Na + and K + Channels A Positive Feedback Cycle Generates the Generate the Action Potential Rising Phase of the Action Potential Rising Phase of Falling Phase of Rest Action Potential Action Potential Open Na + Channels Na + Channels Close; Voltage-Gated Na + Channels K + Channels Open Open Channels Closed Na + + + + - - + + - - Inward I Na Depolarization K + - - + - - + + - - + - - + - - - - + + - - + + + + + + + + + - - - - - - + + + + - - - + - + + + + - - + + + - - + - - + + + + + + - + + - - - - - - - - - + + + - - + - - + + + + + The Voltage Clamp Generates a Depolarizing Step by Voltage Clamp Circuit Injecting Positive Charge into the Axon Command Voltage Clamp: 1) Steps 2) Clamps PNS, Fig 9-2 PNS, Fig 9-2 Opening of Na + Channels Gives Rise to Na + Electronically Generated Current Counterbalances the Na + Membrane Current Influx That Tends to Cause V m to Deviate from Its Commanded Value Command Command g = I/V PNS, Fig 9-2 PNS, Fig 9-2 7
Where Does the Voltage Clamp The Voltage Clamp Interrupts the Interrupt the Positive Feedback Cycle? Positive Feedback Cycle Here Open Na + Open Na + Channels Channels Inward I Na Inward I Na Depolarization Depolarization X 8
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