Ephys Lectures 2 2024 - Cellular Electrophysiology PDF

Membrane potential and action potential membrane potential membrane permeability Lecture #2:1 Goldman-Hodgkin-Katz model NBE-E4120 Cellular Elec...

Membrane potential and action potential membrane potential membrane permeability Lecture #2:1 Goldman-Hodgkin-Katz model NBE-E4120 Cellular Electrophysiology application of GHK equation reversal potential resting potential neuron action potential Determination of the membrane potential ▪ If membrane is permeable to only one ion and there is no electrogenic ionic pumps => Vm is at the Nernst potential for that ion ▪ If membrane is permeable to two ions, in the presence of an impermeant ion => two permeant ions will come to equilibrium with an unequal distribution across the semipermeable membrane ▪ At rest, a typical cell membrane has a much higher permeability to K+ than to Na+, Ca2+ or Cl- => Vrest is close to EK ▪ but it is also permeable to other types of ions ▪ for example, the resting membrane of the squid giant axon is permeable to K+, Cl- and Na+  the resting potential is not equal to EK, ENa or ECl, but is somewhere within these three Ei Typical mammalian neuron 2 Effects of changing - [Cl ]out on Vm a):[𝐾 + ]𝑖𝑛 = 125 [𝐾 + ]𝑜𝑢𝑡 = 2.5 [𝐶𝑙− ]𝑖𝑛 = 2.4 ▪ (a) At rest membrane is permeable to K+ and Cl- [𝐶𝑙− ]𝑜𝑢𝑡 = 120 Vm = EK = ECl = -98.5 mV 30 ▪ (b) after sharp drop in [Cl-]out ECl = -58 log 2.4 = - 63.5 mV. Since EK is still -98.5 mV, Vm will change to -77 mV (in between of two Eeq). ▪ (c) Cl- will not be in electrochemical equilibrium anymore and to restore Donnan equilibrium: drop in [Cl-]out − + − + [𝐾𝑜𝑢𝑡 ] 2.5 𝐶𝑙𝑖𝑛 2.4 [𝐾𝑜𝑢𝑡 ] [𝐶𝑙𝑖𝑛 ] = = 0.02 ≠ = + = − + [𝐾𝑖𝑛 ] 125 − ] [𝐶𝑙𝑜𝑢𝑡 30 [𝐾𝑖𝑛 ] [𝐶𝑙𝑜𝑢𝑡 ] Cl- will diffuse out of the cell untill new Donnan equilibrium: Equilibrium + − + − [𝐾𝑜𝑢𝑡 ][𝐶𝑙𝑜𝑢𝑡 ] = [𝐾𝑖𝑛 ] [𝐶𝑙𝑖𝑛 ] = 2.5 *30 = 75 𝑅𝑇 𝐶𝑜𝑢𝑡 + [𝐾𝑜𝑢𝑡 − ][𝐶𝑙𝑜𝑢𝑡 ] = + [𝐾𝑖𝑛 ] − [𝐶𝑙𝑖𝑛 ] 𝐸𝑒𝑞 = ln Effects of sudden reduction of extracellular [Cl-] on membrane 𝑧𝐹 𝐶𝑖𝑛 potential of isolated frog muscle 3 Effects of changing - [Cl ]out on Vm d):[𝐾 + ]𝑖𝑛 = 125 [𝐾 + ]𝑜𝑢𝑡 = 2.5 [𝐶𝑙− ]𝑖𝑛 = 0.6 [𝐶𝑙− ]𝑜𝑢𝑡 = 30 ▪ (d) [Cl-]in will decrease by ~4 times to 0.6 mM => ECl 30 = -58 log 0.6 = - 98.5 mV = EK 120 ▪ (e) when [Cl-]out = 120 mM, ECl = -58 log 0.6 = - 134 mV but EK = - 98.5 mV and thus Vm will change to - 112 mV (in between of two Eeq) ▪ (f) cell again is not in equilibrium and Cl- flow in drop in [Cl-]out ▪ (g) new equilibrium is achieved when [Cl-]in= 2.4 mM − + 𝐶𝑙𝑖𝑛 2.4 2.5 [𝐾𝑜𝑢𝑡 ] − ] = = = + as in (a) Equilibrium [𝐶𝑙𝑜𝑢𝑡 120 125 [𝐾𝑖𝑛 ] 𝑅𝑇 𝐶𝑜𝑢𝑡 + [𝐾𝑜𝑢𝑡 − ][𝐶𝑙𝑜𝑢𝑡 ] = + [𝐾𝑖𝑛 ] − [𝐶𝑙𝑖𝑛 ] 𝐸𝑒𝑞 = ln Effects of sudden reduction of extracellular [Cl-] (on membrane 𝑧𝐹 𝐶𝑖𝑛 potential of isolated frog muscle 4 Membrane permeability ▪ Membrane permeability P to ion i (cm/s): 𝐽 = −𝑃Δ𝐶  Transmembrane permeability of an ion where J is molar flux (mol/cm2∙s) depends on the diffusion coefficient 𝐷, 𝜕𝐶 water-membrane solubility of ion 𝛽 and Fick’s 1st law: 𝐽𝑑𝑖𝑓𝑓 = −𝐷 thickness of the membrane 𝑙 𝜕𝑥 ▪ Assume homogenous membrane  and based on Stokes-Einstein equation:  C drops linearly in the membrane with distance x: 𝑅𝑇 𝐷=𝑢 𝑑𝐶 Δ𝐶 𝐹 =𝛽⋅ 𝑑𝑥 𝑙 𝑅𝑇 𝑢 ∗ 𝛽 where 𝑙 is membrane thickness 𝑃= and 𝛽 is partition coefficient for ion i 𝐹 𝑙 𝐷∗ 𝛽  permeability depends on ion then 𝐽=− Δ𝐶 mobility in the membrane 𝑢∗ and 𝑙 the temperature T 𝐷∗𝛽 and 𝑃= 𝑙 where ∗ is inside the membrane and D is the diffusion constant for ion i (cm2∙s) Goldman-Hodgkin-Katz model ▪ 𝑅𝑇 𝜕𝐶 𝜕𝑉 Nernst-Planck equation (NPE) is difficult to solve without simplifying: 𝐽 = −𝑢( 𝐹 + 𝑧𝐶 𝜕𝑥 ) 𝜕𝑥 ▪ Goldman, Hodgkin and Katz assumed: 1. ion movement within the membrane obeys Nernst-Planck equation (electrodiffusion) 2. ions move across the membrane independently (no interaction) 3. electric field E in the membrane is constant (i.e. V changes linearly across the membrane of thickness 𝑙) i.e.: 𝜕𝑉 𝑉1 −𝑉2 𝑉𝑚 then 𝑅𝑇 𝜕𝐶 𝑉𝑚 𝐸= 𝜕𝑥 = 𝑙 = 𝑙 𝐽 = −𝑢 −𝑢 𝑧𝐶 𝐹 𝜕𝑥 𝑙 ▪ For the cell at resting state: that are permeable to K+, Na+ and Cl-: 𝐼𝑡𝑜𝑡 = 𝐼𝐾 + 𝐼𝑁𝑎 + 𝐼𝐶𝑙 = 0 𝐼𝑡𝑜𝑡 = ෍ 𝐼𝑖 = 0 𝑖  steady-state membrane potential (i.e. the resting potential) for the cell that are permeable to K+, Na+ and Cl- can be described by Goldman-Hodgkin-Katz voltage equation (GHK): 𝑅𝑇 𝑃𝐾 [𝐾 + ]𝑜𝑢𝑡 + 𝑃𝑁𝑎 [𝑁𝑎+ ]𝑜𝑢𝑡 + 𝑃𝐶𝑙 [𝐶𝑙 − ]𝑖𝑛 𝑉𝑚 = ln also called constant field model 𝐹 𝑃𝐾 [𝐾 + ]𝑖𝑛 + 𝑃𝑁𝑎 [𝑁𝑎+ ]𝑖𝑛 + 𝑃𝐶𝑙 [𝐶𝑙 − ]𝑜𝑢𝑡 where 𝑃𝑖 is relative membrane permeability of ion i see derivation of GHK in Johnston & Wu textbook 6  Goldman-Hodgkin-Katz model ▪ The transmembrane current of the permeant ion is described by Goldman-Hodgkin-Katz current equation: 𝑧𝐹𝑉𝑚 2 2 − 𝑧 𝐹 [𝐶]𝑖𝑛 − [𝐶]𝑜𝑢𝑡 𝑒 𝑅𝑇 𝐼=𝑃 𝑉𝑚 𝑧𝐹𝑉 𝑅𝑇 − 𝑅𝑇𝑚 1−𝑒 ▪ it predicts that membrane current is nonlinear function of membrane potential Outward [𝐶]𝑜𝑢𝑡 ▪ Current-voltage relationship for various [𝐶]𝑖𝑛 ▪ I-V relationship is linear only if [𝐶]𝑜𝑢𝑡 =1 [𝐶]𝑖𝑛 ▪ I-V relation is nonlinear and ionic currents are said to be rectified: ▪ Inward rectification: inward current flows more easily than outward ▪ outward rectification: outward current [𝐶]𝑜𝑢𝑡 flows more easily than inward [𝐶]𝑖𝑛 Inward ▪ a rectifier is an electronic component that passes current preferentially in one direction 7 Example: effects of extracellular [K+] on Vm In the squid giant axon, at rest, the permeability ratios are: pK : pNa : pCl = 1 : 0.03 : 0.1  the membrane is most permeable to K+, less to Cl-, and the least permeable to Na+  Vrest is determined at 20ºC using GHK for [K+]out=10: 1 ⋅ 10 + 0.03 ⋅ 460 + 0.1 ⋅ 40 𝑉𝑟𝑒𝑠𝑡 = 58 log10 mV = −70 mV 1 ⋅ 400 + 0.03 ⋅ 50 + 0.1 ⋅ 540 10 𝐸𝐾 = 58 log10 mV = −93 mV 400 ▪ Vrest by GHK is in agreement with experimental results The effects of extracellular [K+] on Vm. The solid line is As [K+] out increases Vrest is more dependent on K+ given by GHK, dashed line: by Nernst equation. 𝑃𝐾 [𝐾+ ]𝑜𝑢𝑡 +𝑃𝑁𝑎 [𝑁𝑎+ ]𝑜𝑢𝑡 +𝑃𝐶𝑙 [𝐶𝑙 − ]𝑖𝑛 𝑃𝐾 [𝐾+ ]𝑜𝑢𝑡 inside (mM) outside (mM) 𝑉𝑟𝑒𝑠𝑡 = 58 log ≈ 58 log = 𝐸𝐾 K+ 𝑃𝐾 [𝐾+ ]𝑖𝑛 +𝑃𝑁𝑎 [𝑁𝑎+ ]𝑖𝑛 +𝑃𝐶𝑙 [𝐶𝑙 − ]𝑜𝑢𝑡 𝑃𝐾 [𝐾+ ]𝑖𝑛 400 10 Na+ 50 460 Cl- 40 540 => solid and dashed lines coincide 8 Example: effect of pump current on Vm ▪ In fact, Vrest of squid giant axon is a few mV less through the operation of the Na+ - K+ pump ▪ Total current at rest: 𝐼𝑡𝑜𝑡 = 𝐼𝑖 + 𝐼𝑖,𝑝𝑢𝑚𝑝 = 0 where 𝐼𝑖 is a passive current and 𝐼𝑝𝑢𝑚𝑝 is a current generated by pump ▪ Na+ - K+ pump: 3 Na+ out, 2 K+ in => ▪ permeability ratio r = 𝑃𝑁𝑎 / 𝑃𝐾 = 1.5 (i.e. number of Na+ pumped out for each K+ pumped in) 𝐼𝑁𝑎 + 𝐼𝑁𝑎,𝑝𝑢𝑚𝑝 = 0 𝐼𝐾 + 𝐼𝐾,𝑝𝑢𝑚𝑝 = 0 𝑟𝐼𝐾 + 𝐼𝑁𝑎 = 0 𝑟𝐼𝐾,𝑝𝑢𝑚𝑝 + 𝐼𝑁𝑎,𝑝𝑢𝑚𝑝 = 0 𝑟𝑃𝐾 [𝐾 + ]𝑜𝑢𝑡 + 𝑃𝑁𝑎 [𝑁𝑎+ ]𝑜𝑢𝑡 1.5 ⋅ 1 ⋅ 10 + 0.03 ⋅ 440 ⇒ 𝑉𝑟𝑒𝑠𝑡 = 58 log = 58 log10 mV = −88 mV 𝑟𝑃𝐾 [𝐾 + ]𝑖𝑛 + 𝑃𝑁𝑎 [𝑁𝑎+ ]𝑖𝑛 1.5 ⋅ 1 ⋅ 400 + 0.03 ⋅ 50 1⋅10+0.03⋅440 ▪ Without pump (i.e. r =1 ): 𝑉𝑟𝑒𝑠𝑡 = 58 log10 mV = - 82 mV 1⋅400+0.03⋅50 ▪ Effect of pump on Vm is typically less than 15% (more negative due to the pump) Resting membrane potential ▪ Resting membrane potential (Vrest) is relatively static voltage difference across the cell membrane of “electrically inactive” cells ▪ a steady-state condition, i.e. in the absence of external stimuli and opposed to the action potential and graded membrane potential ▪ all open channels and pumps are involved ▪ K⁺ usually having the most influence ▪ Resting membrane potential is determined by the intracellular and extracellular concentrations of ions to which the membrane is permeable and on their permeabilities (GHK equation) 𝑅𝑇 𝑃𝐾 [𝐾 + ]𝑜𝑢𝑡 + 𝑃𝑁𝑎 [𝑁𝑎+ ]𝑜𝑢𝑡 + 𝑃𝐶𝑙 [𝐶𝑙 − ]𝑖𝑛 𝑉𝑟𝑒𝑠𝑡 = ln 𝐹 𝑃𝐾 [𝐾 + ]𝑖𝑛 + 𝑃𝑁𝑎 [𝑁𝑎+ ]𝑖𝑛 + 𝑃𝐶𝑙 [𝐶𝑙 − ]𝑜𝑢𝑡 10 Resting membrane potential Relatively small number of moved charges can cause relatively large changes in the membrane voltage (depending also on Rm) without any relevant change in the ion concentration Example: ▪ to maintain a resting potential of -70 mV, 7 × 10−11 C or about 109 single charged ions are required for a cells with C m of 1 nF (Q = CmV) ▪ This is about a 1/100000 = 0.001% of the total number of ions in a neuron ▪ This charge corresponds to ~ 0.7 nA current for 100 ms (Q = It) => the number of ions moved is negligible when compared to the total number of charges! Terminology: Vm becomes more positive than Vrest, it is said to depolarize Vm returns after depolarization towards Vrest, it is said to repolarize Vm becomes more negative than Vrest, it is said to hyperpolarize 11 Resting membrane potential ▪ Negative Vrest is present in all differentiated cells ▪ Equally rich sets of ion channels exist in both excitable and non-excitable cells ▪ Vrest facilitates the propagation of action potentials in excitable cells ▪ as rapid changes in membrane ion permeabilities lead to a regenerating action potential in nerves ▪ Dynamic membrane potential is critical for many processes in non-excitable cells ▪ cell cycle, cell-volume control, proliferation, muscle contraction (even in the absence of an action potential), wound healing etc. ▪ Different cell types have different Vrest ▪ membranes of neurons and glia contain multiple types of ionic pumps ▪ Na+ -K+ pump make Vm more negative (= hyperpolarized): 3 Na+ out for every 2 K+ in Yang and Brackenbury (2013) 12 Electrical communication Neurons use two distinct signalling modes: ▪ Action potentials (or simply spikes) ▪ all-or-nothing responses: frequency coding (digital signal) ▪ lower information content (bits per s-1) ▪ Na+- and Ca²⁺-dependent APs in excitable cells ▪ regenerated and rapid information transmission over long distances ▪ in myelinated axons propagate without any decrease in amplitude https://en.wikipedia.org ▪ specific threshold (around -45 - -55 mV), cannot summate ▪ Graded potentials ▪ amplitude is directly proportional to the strength of the stimulus: amplitude coding (analog signal) ▪ higher information content ▪ in receptor cells, short interneurons, dendrites and in synapses Responses to light in Responses to stretch in for postsynaptic signal generation photoreceptor mechanoreceptor neurons ▪ information transmitted over short distances ▪ decrease in amplitude as they spread from the point of origin ▪ There are also mixed and intermediate forms ▪ no threshold, can summate of signalling (Ca2+ based AP-like responses) A typical neuron Dendrites with dendritic spines; the receiving part of the neuron with the postsynaptic parts of synapses Graded potentials Soma – the central part The origin locus of action potentials - ”axon hillock” Action potentials Axon – the transmitting part of the neuron: propagation and regeneration of action potentials Action potentials Axon terminal with the synapse, its presynaptic part  Generation of action potential ▪ Hodgkin and Huxley recorded the action potential (AP) in squid giant axon with an intracellular microelectrode: ▪ during the AP, Vm rapidly overshoots 0 mV and approaches ENa ▪ after generation of the AP, Vm repolarizes and becomes more negative than before, generating an afterhyperpolarization  Cole and Curtis: changes in Vm during AP is due to a large increase in conductance 2 ms not known to which ions because ionic currents could not be measured directly The squid giant axon is a projection of a single neuron that known: AP depends on the presence of controls water propulsion. It is a up to 1.5 mm in diameter extracellular Na+ and several cm long. http://www.mun.ca/biology/ proposed: AP is generated through a Intracellular recording of action potential generation in the squid giant axon. rapid increase gNa 15 Increase in Na+ and K+ conductance generates APs ▪ Development of voltage-clamp technique by Cole in 1949: possible to record Na+ and K+ currents underlying the action potential Voltage-clamp analysis by Hodgkin & Huxley: 1. Increasing the voltage from -60 to 0 mV produces a transient flow of positive charge into the cell (an inward current) 2. This inward current followed by a sustained flow of positive charge out of the cell (an outward current)  Hodgkin & Huxley: the transient inward current is carried by Na+ ions and the sustained outward current is mediated by K+ ions: ▪ Na+ in seawater is replaced by choline (which does not pass through Na+ channels), then only the outward current, which corresponds to IK ▪ Subtracting IK from the recording in normal seawater gives the inward Na+ current, INa Voltage-clamp analysis reveals the ion currents ▪ IK activates more slowly than INa; INa inactivates within several ms underlying action potential generation 16 Increase in Na+ and K+ conductance generates APs ▪ Ionic currents can also be isolated using the pharmacological blockers: Protocol: voltage steps from -45 to +75 mV in 15-mV steps In control: the amplitude-time course of the inward Na+ and outward K+ currents Selective block of INa with tetrodotoxin (TTX): only IK Selective block of IK with tetraethylammonium (TEA): only INa  differences between Na+ and the K+ currents  Na+ current both rapidly activates and inactivates, whereas the K+ current only slowly activates In 1952, Hodgkin and Huxley developed theory of the time and voltage dependencies of Na+ and K+ currents in the squid giant axon Voltage-clamp analysis reveals the ionic currents underlying action potential generation 17 Video: The squid and its giant nerve fibre Homework #2 see squid with your own eyes: https://www.youtube.com/watch?v=g 2hysrWbuZs 18  Electrical properties of excitable membranes equivalent circuit representation Lecture #2:2 membrane and ionic conductances parallel conductance model current-voltage relationship Equivalent circuit representation ▪ Biological membranes behave electrically like ▪ Total transmembrane current has resistive resistances in parallel with capacitances and capacitive components: 𝑑𝑉𝑚 𝐼𝑚 = 𝐼𝑖 + 𝐼𝐶 = 𝐺𝑚 (𝑉𝑚 − 𝑉𝑟𝑒𝑠𝑡 ) + 𝐶𝑚 ▪ Membrane specific capacitance 𝐶𝑚  1 F/cm2 𝑑𝑡 ▪ Membrane specific resistance 𝑅𝑚 , [𝑅𝑚 ] = ·сm2 ▪ Ionic current is a product of total membrane conductance and its driving ▪ membrane specific conductance 𝐺𝑚 = 1/ 𝑅𝑚 , [𝐺𝑚 ] = S·сm-2 force: ▪ represent ion permeation through ion channels 𝐼𝑖 = 𝐺𝑚 (𝑉𝑚 − 𝑉𝑟𝑒𝑠𝑡 ) ▪ in excitable cells 𝑅𝑚 is highly dependent on Vm and t ▪ Driving force is the difference between membrane potential and resting/reversal potential: 𝑉𝑚 − 𝑉𝑟𝑒𝑣,𝑖 𝐸𝑟𝑒𝑠𝑡 Equivalent electrical circuit of the cell membrane 20 Membrane potential: summary ▪ Reversal potential (Vrev) is the membrane 0 ▪ Equilibrium potential of an ion (Nernst potential at which the net flow of a specific ion or ෍ 𝐼𝑖 = 0 potential) is the membrane potential at a combination of ions through channels is zero 𝑖 which there is no net movement of a ▪ i.e. when the direction of ionic current reverses = the specific ion across membrane inward and outward currents of the ions are exactly balance each other ▪ it is a property of a specific ion! ▪ it is a property of channels! ▪ If the channels conduct only one type of ion and and no ionic pumps are operating, then Vrev and equilibrium ▪ Resting membrane potential (Vrest) is potential Eeq coincide: voltage difference across the cell 𝑅𝑇 [𝐶]𝑜𝑢𝑡 membrane in a steady-state condition 𝑉𝑟𝑒𝑣 = 𝐸𝑒𝑞,𝑖 = ln 𝑧𝐹 [𝐶]𝑖𝑛 ▪ with zero net transmembrane current: ▪ If the channels are permeable to more than one type of ion: 𝐼𝑚 = ∑𝑔𝑖 (𝑉, 𝑡)(𝑉𝑚 − 𝑉𝑟𝑒𝑠𝑡 ) = 0 then Vrev is a resting potential and a weighted average of the equilibrium potentials of the permeable ions: ▪ it is a property of entire cell ∑𝑖 𝑔𝑖 (𝑉, 𝑡)𝐸𝑒𝑞,𝑖 𝑔𝐾 𝐸𝐾 + 𝑔𝑁𝑎 𝐸𝑁𝑎 + 𝑔𝐶𝑙 𝐸𝐶𝑙 membrane! 𝑉𝑟𝑒𝑣 = 𝑉𝑟𝑒𝑠𝑡 = 𝑉𝑟𝑒𝑣 = 𝑉𝑟𝑒𝑠𝑡 = ▪ GHK voltage equation ∑𝑖 𝑔𝑖 (𝑉, 𝑡) 𝑔𝐾 + 𝑔𝑁𝑎 + 𝑔𝐶𝑙 parallel conductance model 21 Membrane and ionic conductances Membrane conductance 𝐺𝑚 is the sum of all ionic ▪ For passive individual ionic currents 𝐼𝑖 conductanes 𝑔𝑖 in the membrane (for each ion type): ▪ i.e. ions flow down its electrochemical gradient without active transport: 𝐺𝑚 = ෍ 𝑔𝑖 𝑅𝑇 𝐶𝑜𝑢𝑡 𝑖 𝐼𝑖 = 𝑔𝑖 (𝑉𝑚 − 𝐸𝑒𝑞,𝑖 ) 𝐸𝑒𝑞,𝑖 = ln 𝑧𝐹 𝐶𝑖𝑛 Linear membrane: When 𝑉𝑚 = 𝐸𝑒𝑞,𝑖 , 𝐼𝑖 = 𝑓 𝑉𝑚 , 𝐸𝑒𝑞,𝑖 , 𝑡 = 0  exhibits a linear relation between membrane ionic current > > and transmembrane potential: When 𝑉𝑚 𝐸𝑒𝑞,𝑖 , 𝐼𝑖 = 𝑓(𝑉𝑚 , 𝐸𝑒𝑞,𝑖 , 𝑡) 0 < < 𝐼𝑖 = 𝐺𝑚 (𝑉𝑚 − 𝐸𝑒𝑞,𝑖 ) ▪ holds also when several types of passive conductances present where 𝐺𝑚 = const and I-V relation is a straight line Non-linear membrane:  may vary from voltage and time i.e. 𝐼𝑖 = 𝑓(𝑉𝑚 , t) 𝐼𝑖 (𝑉𝑚 , 𝑡) = 𝐺𝑚 (𝑉𝑚 − 𝐸𝑒𝑞,𝑖 ) linear range non-linear range 22 Parallel conductance model  The equivalent circuit of excitable membrane consists of a capacitor (𝐶𝑚 ) in parallel with three pathways, each consisting of a battery (𝐸𝑒𝑞 ) in series with an ionic conductance (𝑔𝑖 )  Total membrane current mediated by K+, Na+ and Cl-: parallel conductance model 𝐼𝑡𝑜𝑡 = 𝐼𝐶 + 𝐼𝐾 + 𝐼𝑁𝑎 + 𝐼𝐶𝑙 60 𝑑𝑉𝑚 𝐼𝑡𝑜𝑡 = 𝐶𝑚 + ෍ 𝑔𝑖 (𝑉𝑚 − 𝐸𝑖 ) 40 𝑑𝑡 𝑖 20 𝑑𝑉𝑚 0 𝐼𝑡𝑜𝑡 = 𝐶𝑚 + 𝑔𝐾 (𝑉𝑚 − 𝐸𝐾 ) + 𝑔𝑁𝑎 (𝑉𝑚 − 𝐸𝑁𝑎 ) + 𝑔𝐶𝑙 (𝑉𝑚 − 𝐸𝐶𝑙 ) 𝑑𝑡 -20 𝑑𝑉𝑚  when 𝑑𝑡 =0, i.e. at rest or at the AP peak: -40 -60 𝑔𝑁𝑎 𝐸𝑁𝑎 + 𝑔𝐾 𝐸𝐾 + 𝑔𝐶𝑙 𝐸𝐶𝑙 -80 𝑉𝑚 = 𝑔𝑁𝑎 + 𝑔𝐾 + 𝑔𝐶𝑙 Current-voltage relations ▪ I/V curve is the most common way to analyze membrane conductances linear range ▪ usually non-linear ▪ obtained by presenting a square voltage pulses of increasing amplitude to a voltage- non-linear range clamped cell and recording the current responses ▪ for each trace, the peak (maximum) current is measured and plotted as a function of the voltage ▪ Different currents can be isolated by using blockers, specially designed solution or voltage protocols Potassium currents (left) and the I/V curve (right) Current-voltage relations Isolation of different Ca2+ currents using blocker nifedipine ▪ nifedipine inhibits L-type channels, same protocol as in control => this is T- type channel (left) ▪ => subtracting this curve from the first curve gives the I-V curve for the L- 𝑉𝑟𝑒𝑣 type channel (right) The whole-cell current is proportional to both the fraction of open channels and the driving force I/V curve of a total (L- and T-types) => the fractional conductance g/gmax can be obtained by dividing the current calcium current amplitude by the driving force and normalizing with respect to the maximum conductance nifedipine => T-type subtracted g/gmax 𝐼𝑖 = 𝐺𝑚 (𝑉𝑚 − 𝑉𝑟𝑒𝑣 ) Activation curves of T and L types of calcium channels Segregated I/V curves of the type T (left) and type L (right) channel