Michale Fee Methods in Computational Neuroscience 2015 Marine Biological Laboratory Woods Hole MA Brief Introduction to Neuronal Biophysics

Neuronal biophysics We are going to build a biophysical model of a neuron. Hodgkin and Huxley, 1938 A conceptual model based on simple concepts from electrical circuits A mathematical model that we can use to calculate properties of neurons 2

A mathematical model of a neuron Equivalent circuit model g Na gKgK gLgL ELEL EKEK E Na IeIe VmVm C Alan Hodgkin Andrew Huxley,

What are the conductors in a neuron? Water is a polar solvent Intracellular and extracellular space is filled with salt solution (~100mM) –6x10 19 ions per cm 3 (25Å spacing) Na + H+H+ O-O- H+H+ O-O- H+H+ H+H+ O-O- H+H+ H+H+ Cl - H+H+ O-O- H+H+ O-O- H+H+ H+H+ O-O- H+H+ H+H+ 1Å 4 Current flows through a salt solution by the movement of ions in an applied electric field + - V I L

Ohm’s Law + - V I Surface area A L What happens to current if we separate the electrodes? If we make them with larger area? The relation between voltage (V) and current (I) is just given by Ohm’s Law where –V is voltage (Volts, V) –I is current (Amperes, A) –R is resistance (Ohms, Ω) 5 Resistance of a conductive medium is given by resistivity

Ohm’s Law + - V I Surface area A L The relation between voltage (V) and current (I) is just given by Ohm’s Law where –V is voltage (Volts, V) –I is current (Amperes, A) –R is resistance (Ohms, Ω) 6 resistivity ρ = ~60 Ω  cm for mammalian saline – the brain has lousy conductors… Compare to ρ = 1.6 μΩ  cm for copper

Phospholipid bilayer: polar head non-polar tail intracellular extracellular IeIe C V m = V in -V out Equivalent circuit VmVm Why is this a capacitor? A capacitor is two conductors separated by an insulator A neuron is a capacitor 7

intracellular extracellular IeIe C VmVm IcIc Q: charge (Coulombs, C = 6 x charges) C: capacitance (Farads, F) V: voltage difference across capacitor (Volts, V) Q: charge (Coulombs, C = 6 x charges) C: capacitance (Farads, F) V: voltage difference across capacitor (Volts, V) But, Kirchoff’s current law tells us that the sum of all currents into a node is zero Definition of capacitive current Thus, we can write the differential equation that describes the change in voltage of a capacitor with injected current I e has units of Amperes, which is Coulombs per second A neuron is a capacitor 8

We can integrate this differential equation over time, starting with initial voltage V 0 at time zero. Think about the integral as adding up all the current from time 0 to time t Thus, the total change in voltage is just given by Response of a neuron to injected current 9 capacitor

Some examples 0 0 VmVm time I0I0 0 I0I0 0 VmVm

Why are are we interested in understanding the response of neurons to injected current? –Because, in the brain, neurons have current injected to them: From other neurons (through synapses) Or as a result of sensory stimuli The response of a neuron to injected current is an important neuronal computation: A neuron is able to perform analog integration of its inputs over time! 11

Membrane currents and ion channels 12 Start by analyzing the simplest membrane ionic current – a leak conductance! We can model this as a resistor

Our equation for our model becomes: electrode current membrane capacitive current membrane ionic current outward current ‘+’ leaving the cell ⇒ positive inward current ‘+’ entering the cell ⇒ negative C VmVm 13 intracellular extracellular

Simple case: a leak Membrane currents can be modeled using Ohm’s Law Plugging this into our equation above, we get 14

Multiplying by R L, we get: What is the steady-state solution to this equation? Set dV/dt = 0 We find that: Thus, we can rewrite our equation as follows 15

We can rewrite our equation in the following form: We see that the derivative is negative if V>V ∞ positive if V<V ∞ Thus, the voltage constantly approaches the value V ∞ V(t) time V0V0 V∞V∞ An aside about first-order linear differential equations And it approaches at a rate proportional to how far V is from V ∞ V(t) time V0V0 V∞V∞ small τ What kind of function is this? τ 16

Under the condition that I e is constant (and thus V ∞ is constant): Even though this solution applies only in the case of constant V ∞ ….. it is very useful anyway! 17

Response to current injection time 18

Origin of 10 millisecond time scale 19

We have described the relation between voltage and current using Ohms Law A closer look at membrane resistance R L has units of Ohms (Ω) G L has units of Ohms -1 or Siemens (S) We can rewrite Ohm’s Law in terms of a quantity called ‘conductance.’ I-V curve 20

Conductances in parallel add Twice the area, twice the holes, twice the conductance, twice the current at a given voltage Membrane area (mm 2 ) Specific leak conductance (mS/mm 2 ) I1I1 I2I2 V + 21

Capacitances in parallel add! Thus, the capacitance of a cell depends linearly on surface area membrane area specific capacitance (10 nF/mm 2 ) Capacitance (nF) A closer look at membrane capacitance 22

Membrane time constant Thus, the time constant of a neuron is a property of the membrane, not dependent on cell geometry (size, shape, etc!). 23

Neurons have batteries Where do the ‘batteries’ of a neuron come from? 24

Neurons have batteries Lets imagine a beaker filled with water and separated into two compartments with a thin membrane. Now pour KCl into the left side. Where do the ‘batteries’ of a neuron come from? Now poke a hole in the membrane and see what happens. By ‘hole’ I mean a ‘non-selective’ pore 25

Neurons have batteries time [K] [K] out [K] in [K] 0 [K] ∞ =[K] 0 /2 ‘Non-selective’ pore passes all ions 26

Neurons have batteries Why do the ions stop flowing from side 1 to side 2? [K] [K] out [K] in [K] 0 time ‘Ion-selective’ pore passes only K + ions I K (t) time 27

Neurons have batteries ‘Ion-selective’ pore passes only K + ions I K (t) time ΔV=V 1 -V 2 time 0 The voltage difference changes in a direction that opposes the flow of ions. EKEK It reaches an ‘equilibrium potential’ at a value that gives zero net flow of ionic current. This voltage difference is a battery for our model neuron!! 28

Nernst Potential We can compute the equilibrium potential using the Boltzmann equation: 29 Nernst potential for K ion

IonCytoplasm (mM) Extracellular (mM) Nernst (mV) K+K The Nernst potential for potassium (K) Intracellular and extracellular concentrations of ionic species, and the Nernst potential = 25mV at 300K (room temp) for monovalent ion 30

The Nernst potential is different for different ions IonCytoplasm (mM) Extracellular (mM) Nernst (mV) K+K Na Cl Ca Intracellular and extracellular concentrations of ionic species, and the Nernst potential 31

EKEK What does the I-V curve look like now? We model this as a conductance (or resistance) in series with a battery. A neuron is a leaky capacitor with batteries 32

Our equation is now: IeIe C V GKGK IcIc IKIK EKEK + 33

Response to current injection time 34

A mathematical model of a neuron g Na gKgK gLgL ELEL EKEK E Na IeIe VmVm C ms Spike generator Spike generator 35

Integrate and Fire Model All spikes are the same. (No information carried in the details of action potential waveforms.) While APs (spikes) are important, they are not what neurons spend most of their time doing. Spikes are very fast (~1ms in duration). This is much shorter than the typical interval between spikes. Most of the time, a neuron is ‘integrating’ its inputs. (Separation of timescales) 250 ms 36

Integrate and fire model of a neuron GLGL ELEL IeIe VmVm C + Spike generator Spike generator The spike generator is very simple. When the voltage reaches the threshold V th, it resets the neuron to a hyper- polarized voltage V res. Louis Lapique, 1907 Membrane potential (mV) 100 ms Mooney, nA 37

Integrate and fire IeIe V C GLGL ELEL + Spike generator Spike generator Let’s calculate the firing rate of our neuron We’ll first consider the case where there is no leak. V th V res 38

Integrate and fire IeIe V C Spike generator Spike generator Let’s calculate the firing rate of our neuron We’ll first consider the case where there is no leak. V th V res 39

Integrate and fire IeIe V C GKGK EKEK + Spike generator Spike generator V th V res Now we’ll put a K- conductance in now. V∞V∞ EKEK 40 What happens when

Integrate and fire with leak V th EKEK V∞V∞ What happens just at threshold? 41 The time to reach threshold ( ) is: very long very sensitive to injected current f.r. 0 This is called the rheobase.

Integrate and fire with leak V th V res V∞V∞ 42

At high input currents, the solution has a simple approximation Integrate and fire 43 This equation is linear in injected current I e, just like the case of no leak! The F-I curve of many neurons look approximately like this! f.r. 0

A mathematical model of a neuron Alan Hodgkin Andrew Huxley, 1952 g Na gKgK gLgL ELEL EKEK E Na IeIe V C Spike generator Spike generator 44

Total membrane current is the sum over the currents from all ionic species i. Outline of HH model This is the total membrane ionic conductance, and it includes the contribution from sodium channels, potassium channels and a ‘leak’ conductance. g Na gKgK gLgL ELEL EKEK E Na IeIe V C The equation for our HH model neuron is 45

Outline of HH model g Na gKgK gLgL ELEL EKEK E Na IeIe V C The sodium conductance is time-dependent and voltage-dependent The potassium conductance is time-dependent and voltage-dependent The leak conductance is neither time-dependent nor voltage-dependent 46

Outline of HH model g Na gKgK gLgL ELEL EKEK E Na IeIe VmVm C Voltage and time-dependent ion channels are the ‘knobs’ that control membrane potential. Na-channels push the membrane potential positive. K-channels push the membrane potential negative. Together these channels generate an action potential. 47

Voltage and Time dependence H&H studied the properties of K and Na channels in the squid giant axon. In particular they studied the voltage and time dependence of the conductance. 1mm diameter! Hodgkin and Huxley, 1938 Most axons in our brain are 1um dia 48

This not easy! Because as soon as you depolarize the axon, the axon begins to spike! The best way study the time and voltage dependence of ionic currents is to suddenly ‘set’ the voltage at different values and measure the current required to hold that voltage. Then plot the I-V curve. 49 Voltage Clamp

RmRm + - VcVc V out VmVm IeIe RiRi The basic equation of an op-amp is: where G is the gain, typically ~10 5 or 10 6, so that even a fraction of a millivolt difference between the positive and negative inputs makes the output swing to its voltage limits (typically ±10V ). The key component is an operational amplifier (op- amp) + - V+V+ V-V- A voltage clamp is a device that holds the membrane potential of a cell to any desired ‘command’ voltage V c, and measures the current required to hold that voltage. G is gain here, not conductance! 50

Voltage Clamp IeIe RmRm + - VcVc V out VmVm RiRi + - VcVc VmVm RiRi RmRm IeIe If V m is below V c  V out is large and positive. –This drives current into the neuron. If V m is above V c  V out is large and negative –This pulls current out of the neuron This is called ‘negative feedback’. 51

Voltage Clamp IeIe RmRm + - VcVc V out VmVm RiRi This is called ‘negative feedback’. 52 It is easy to show that, for large G: Thus, the voltage clamp circuit drives whatever current ( I e ) is necessary to ‘clamp’ the voltage of the neuron to the command voltage. During a voltage clamp experiment, we step the V c around within the voltage range of interest and measure I e. By measuring the voltage drop across R i. How do we measure I e ?

Ionic currents + - VcVc VmVm IeIe 53

Ionic currents How do we figure out the contribution of Na and the contribution of K? Ionic substitution (e.g. replace extracellular NaCl with choline chloride) 54 K current Na current

-80 mV -40 mV 0 mV 40 mV -80 mV -40 mV 0 mV 40 mV K current Na current Ionic currents 55

Ionic currents K current Na current 56

Ionic currents (voltage dependence) We used the voltage clamp to measure current as a function of voltage. But what we are really trying to extract is conductance as a function of voltage! 57

Ionic currents (Voltage dependence) 58

Ionic currents (Voltage dependence) Maximal conductance Sigmoidal voltage-dependence of activation Maximal conductance 59

Ionic currents (Time dependence) Time after start of pulse (ms) activation Time after start of pulse (ms) activation inactivation 60

Ionic currents (time and voltage dependence) Time after start of pulse (ms) Specific conductance (full scale =20 mS/cm 2 ) Specific conductance (full scale =20 mS/cm 2 ) 61

Each of these shows the current on a single trial Average across all the trials ‘ensemble average’ Single channel K-current Time after start of pulse (ms) Single Channels So far, we have been discussing total currents into a neuron. However, these total currents result from ionic flow through thousands of individual ion channels. It is possible to record from single ion channels using a ‘patch clamp’. VmVm + - VcVc IeIe Neher & Sakmann Single channel Na-current Time after start of pulse (ms) 62

Individual ion channels are either OPEN or CLOSED. Unitary ‘open’ conductance Total open ionic conductance Ionic conductance in terms of single channels The total conductance through a membrane is given by the total number of open channels times the unitary conductance of the ion channel ( ). Number of open channels is just the total number of channels times the probability of any channel being open: 63

How do we describe the probability that a channel is open? Each subunit has a voltage sensor and gate to turn the channel on and off. subunit pore The pore of a K + channel is formed by 4 identical subunits. Lets start with the K channel. Ionic conductance in terms of single channels Each subunit has two states: ‘open’ and ‘closed’. ‘n’ is the probability that a subunit is open. Finally, all subunits must be in the ‘open’ state for the channel to be permeable. 64

How do we describe the probability that a channel is open? subunit pore Assuming independence Ionic conductance in terms of single channels If ‘n’ is the probability that one subunit is open, then the probability that all four subunits is open is given by: H & H called the ‘gating variable’ for the potassium current And we can write the K current as: We can now write down the conductance of our K channels as: is the maximal open conductance 65

Ion selective pore Gating charges (voltage sensor) Lipid bilayer Gate OPEN CLOSED Ionic conductance in terms of single channels Many ion channels (including Na and K) are voltage dependent. Intracellular Extracellular One can derive the voltage-dependence based on first principles using the Boltzmann equation…! 66

Note that for any given subunit or channel, when we change the voltage, the energy levels shift (nearly) instantaneously. The probability of making a transition changes instantly, however the number of open channels does not change immediately. We have to wait for thermal fluctuations to kick the channel or subunit open (i.e. for the subunit to undergo a conformational change). This takes time! We are going to model the transitions between open and closed states with a simple rate equation. We can do this because we have many channels to average over. Probability per unit time; units are 1/s Now get the time dependence! 67

Change in the number of open subunits The number of closed subunits that open The number of open subunits that close The number of closed subunits (1-n) X the probability that a closed subunit opens per unit time (α n ) The number of open subunits (n) X the probability that an open subunit closes per unit time (β n ) Change in the number of open subunits per unit time 68

Thus, we can rewrite this equation in terms of the steady state open probability and a time constant: Let’s rewrite this equation as follows: The steady state solution! A time constant! are all voltage dependent Remember… Remember: n is the probability that a subunit is open. 69

Response to voltage change 70 How does the gating variable ‘n’ change as we step the membrane potential? time activationdisactivation

71 time How does the conductance change as we step the membrane potential? The shape of the K conductance was well fit by a rising exponential raised to the fourth power. H & H inferred from this that the K-current was governed by four independent first-order processes! (They didn’t know about the structure of K-channels at the time!) Response to voltage change Time after start of pulse (ms)

72 Measuring the parameters time By measuring the persistent conductance at different voltages, H&H were able to measure n ∞ as a function of voltage. Membrane potential (mV) By measuring the time course of the conductance at onset and offset of the voltage steps, they were able to measure τ n as a function of voltage. Membrane potential (mV) Time constant (ms)

Hodgkin and Huxley summarized their data using algebraic expressions for the rate functions Measuring the parameters Remember that… 73

Why did we do all of this? Measuring the parameters Because once we have expressions for, we can integrate the differential equation for : to get the potassium conductance as a function of V and t: and the potassium current as a function of V and t: 74

Before we move on the the Na current, let’s see how we put all this together in our model neuron. Here are the steps: Overview of the big picture Integrate one time step to get Compute Compute K current: Compute total membrane current: Integrate to get at next time step Start with at time step t Compute 75

Transient conductances (sodium) Na and K conductances differ an important way: Time after start of pulse (ms) Specific conductance (full scale =20 mS/cm 2 ) Specific conductance (full scale =20 mS/cm 2 ) activation inactivation persistence Time after start of pulse (ms) 76

77 time H & H modeled the sodium conductance using a similar mechanism for activation. A new gating variable ‘m’. The rising phase of the Na current is well fit by a rising exponential raised to the third power. The sodium conductance We can measure the parameters

78 Measuring the parameters time By measuring the peak Na conductance at different voltages, H&H were able to measure m ∞ as a function of voltage. Membrane potential (mV) By measuring the time course of the conductance at onset and offset of the voltage steps, they were able to measure τ m as a function of voltage. milliseconds Membrane potential (mV) Time constant (ms) Note the similarity to K-channel activation…

Sodium channel inactivation CLOSED (intactivated) + IntracellularExtracellular Inactivation particle OPEN + HH postulated an additional voltage-dependent inactivation gate. 79

Sodium channel inactivation Dynamics of inactivation are captured by a new gating variable ‘h’. Specific conductance (full scale =20 mS/cm 2 ) time inactivation disinactivation 80

81 Measuring the parameters How do we measure inactivation and recovery from inactivation? Membrane potential (mV) milliseconds Membrane potential (mV) Time constant (ms) 1.Hold V m at different values 2.Let the Na channels inactivate 3.Then measure the Na current! holding potential measurement step

Measuring the parameters Again… Just as for n and m, HH wrote down simple arithmetic expressions that fit their measurements of the parameters. 82

Putting our two Na-channel gating variables together, we get: Assumes independence The sodium conductance is: And the sodium current is: The sodium conductance The probability of having a Na channel open is: NOT ! But it’s not so bad 83

Putting it all together! Start with initial contition at time step t 0 Compute Compute : Total membrane current 84

Putting it all together! 85

A few extra slides

Voltage Clamp We can show that the very high gain of the op-amp makes it able to control the membrane potential to match the command voltage. + - VcVc V out VmVm RiRi RmRm IeIe Note that the series resistors form a voltage divider VmVm V out RiRi RmRm 87

Voltage Clamp Thus, for large G: Solve for V m : IeIe RmRm + - VcVc V out VmVm RiRi 88