Brief Introduction to Neuronal Biophysics Michale Fee Methods in Computational Neuroscience 2017 Marine Biological Laboratory Woods Hole MA

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

A mathematical model of a neuron Equivalent circuit model gNa gK gL EL EK ENa Ie Vm C + Alan Hodgkin Andrew Huxley, 1952

What are the conductors in a neuron? Water is a polar solvent Intracellular and extracellular space is filled with salt solution (~100mM) 6x1019 ions per cm3 (25Å spacing) Na+ H+ O- Cl- 1Å Current flows through a salt solution by the movement of ions in an applied electric field + - V I L

Ohm’s Law + - V I 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) Surface area A L What happens to current if we separate the electrodes? If we make them with larger area? Resistance of a conductive medium is given by resistivity Ask questions

Ohm’s Law + - V I 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) Surface area A L resistivity Ask questions for mammalian saline – the brain has lousy conductors… Compare to for copper

A neuron is a capacitor Why is this a capacitor? Phospholipid bilayer: polar head non-polar tail Why is this a capacitor? intracellular extracellular Ie C Vm = Vin-Vout Equivalent circuit — two conductors separated by an insulator Typical electric fields across the bilayer are 250kV/cm Vm

A neuron is a capacitor + + - + - + - + - - Ic Vm Ie Ic intracellular Charge imbalance: Voltage difference: Ic + - + - Vm + - + - - C Ie Ic extracellular Draw charges inside and outside the cell membrane As positive charges build up on the inside of the membrane, they repel positive charges away from the outside of the membrane… Q: charge (Coulombs, C = 6 x 1018 charges) C: capacitance (Farads, F) V: voltage difference across capacitor (Volts, V) This looks like a current flowing through the capacitor! 8

A neuron is a capacitor Ic Vm Ie intracellular extracellular Ie C Vm Ic Definition of capacitive current But, Kirchoff’s current law tells us that the sum of all currents into a node is zero Thus, we can write the differential equation that describes the change in voltage of our neural capacitor with injected current Ie has units of Amperes, which is Coulombs per second 9

Response of a neuron to injected current capacitor We can integrate this differential equation over time, starting with initial voltage V0 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

Some examples Vm time I0 I0 Vm time

Why are are we interested in understanding the voltage response of neurons to injected current? Because, in the brain, neurons have current injected to them And because many neuronal processes depend on voltage Thus voltage and current are key dynamical variables 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!

Membrane currents and ion channels Start by analyzing the simplest membrane ionic current – a leak conductance! We can model this as a resistance between intracellular an extracellular space

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

Simple case: a leak Membrane currents can be modeled using Ohm’s Law Draw I-V curve for leak case Plugging this into our equation above, we get

Multiplying by RL, 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

An aside about first-order linear differential equations 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∞ And it approaches at a rate proportional to how far V is from V∞ V(t) time V0 V∞ V(t) time V0 V∞ small τ What kind of function is this? τ τ

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

Response to current injection time

What is the neuronal time scale?

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

Conductances in parallel add V + I1 I2 Twice the area, twice the holes, twice the conductance, twice the current at a given voltage Membrane area (mm2) Specific leak conductance (mS/mm2)

A closer look at membrane capacitance Capacitances in parallel add! Thus, the capacitance of a cell depends linearly on surface area membrane area specific capacitance (10 nF/mm2) Capacitance (nF)

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

Live neurons have… batteries! Where do the ‘batteries’ of a neuron come from?

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

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

Neurons have batteries ‘Ion-selective’ pore passes only K+ ions [K] [K]out [K]in [K]0 time IK (t) time ΔV=V1-V2 time Why does the current stop flowing? EK Equilibrium potential that gives zero net flow of ionic current.

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

The Nernst potential for potassium (K) Intracellular and extracellular concentrations of ionic species, and the Nernst potential Ion Cytoplasm (mM) Extracellular (mM) Nernst (mV) K+ 400 20 -75 = 25mV at 300K (room temp) for monovalent ion 30

The Nernst potential is different for different ions Intracellular and extracellular concentrations of ionic species, and the Nernst potential Ion Cytoplasm (mM) Extracellular (mM) Nernst (mV) K+ 400 20 -75 Na+ 50 440 +55 Cl- 52 560 -60 Ca++ 10-4 2 +125

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

IK Ic GK V C Ie EK + Our equation is now:

Response to current injection time

A mathematical model of a neuron Spike generator gNa gK gL EL EK ENa Ie Vm C + 250 ms

Integrate and fire model of a neuron 250 ms 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) All spikes are the same. (No information carried in the details of action potential waveforms.)

Integrate and fire model of a neuron GK EK Ie Vm C + Spike generator The spike generator is very simple. When the voltage reaches the threshold Vth, it resets the neuron to a hyper-polarized voltage Vres. Louis Lapique, 1907 Vth Vres Vm(t) time

Integrate and fire model of a neuron Let’s calculate the firing rate of our neuron GK EK + We’ll first consider the case where there is no leak. Spike generator V C Ie Vth Vres

Integrate and fire model of a neuron Let’s calculate the firing rate of our neuron We’ll first consider the case where there is no leak. Spike generator V C Ie Vth Vres

Integrate and fire model of a neuron Now we’ll put a K-conductance in now. Ie V C GK EK + Spike generator V∞ Vth Vres EK What happens when

Integrate and fire with leak What current does it take to reach threshold? V∞ Vth This is called the rheobase. EK f.r. The time to reach threshold ( ) is: very long very sensitive to injected current

Integrate and fire with leak Vth Vres V∞

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

A mathematical model of a neuron Spike generator gNa gK gL EL EK ENa Ie V C + Alan Hodgkin Andrew Huxley, 1952

Outline of HH model gNa gK gL EL EK ENa Ie V C + Total membrane current is the sum over the currents from all ionic species i. This is the total membrane ionic conductance, and it includes the contribution from sodium channels, potassium channels and a ‘leak’ conductance. The equation for our HH model neuron is

Outline of HH model V Ie gNa gK gL C ENa EK EL + 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

Outline of HH model Voltage and time-dependent ion channels are the ‘knobs’ that control membrane potential. gNa gK gL EL EK ENa Ie Vm C + Na-channels push the membrane potential positive. K-channels push the membrane potential negative. Together these channels generate an action potential.

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

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

Voltage Clamp A voltage clamp is a device that holds the membrane potential of a cell to any desired ‘command’ voltage Vc, and injects the current required to hold that voltage. The key component is an operational amplifier (op-amp) + - V+ V- Vc + Vout - Ie Ri Vm The basic equation of an op-amp is: where G is the gain, typically ~105 or 106, 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 ). Rm G is gain here, not conductance!

Voltage Clamp - - Vc Vc It is easy to show that, for large G: + - Vc Vout Vm Ri Rm Ie Ie Rm + - Vc Vout Vm Ri If Vm is below Vc then Vout is large, positive. This drives current into the neuron. It is easy to show that, for large G: During a voltage clamp experiment, we step the Vc around within the voltage range of interest and measure Ie .

HH ionic currents + - Vc Vm Ie

HH 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) K current Na current

HH ionic currents K current Na current 40 mV 40 mV 0 mV 0 mV -40 mV

HH ionic currents K current Na current

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!

Ionic currents (Voltage dependence)

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

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

(time and voltage dependence) Ionic currents (time and voltage dependence) Specific conductance (full scale =20 mS/cm2) Specific conductance (full scale =20 mS/cm2) 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. Single channel K-current Time after start of pulse (ms) Single channel Na-current Time after start of pulse (ms) It is possible to record from single ion channels using a ‘patch clamp’. Vm + - Vc Ie Erwin Neher & Bert Sakmann 70’s and 80’s Each of these shows the current on a single trial Average across all the trials: ‘ensemble average’

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

Ionic conductance in terms of single channels How do we describe the probability that a channel is open? Lets start with the K channel. subunit pore The pore of a K+ channel is formed by 4 identical subunits. Each subunit has a voltage sensor and gate to turn the channel on and off. Each subunit has two states: ‘open’ and ‘closed’. ‘n’ is the probability that a subunit is open. R. Mackinnon, 1998 Finally, all subunits must be in the ‘open’ state for the channel to be permeable.

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

Ionic conductance in terms of single channels Many ion channels (including Na and K) are voltage dependent. OPEN CLOSED Gating charges (voltage sensor) + + + + + + + + + + + + Gate Ion selective pore Lipid bilayer Intracellular Extracellular One can derive the voltage-dependence based on first principles using the Boltzmann equation…! 65

Now get the time dependence! 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! Probability per unit time; units are 1/s 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.

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 (αn) The number of open subunits (n) the probability that an open subunit closes per unit time (βn) Change in the number of open subunits per unit time Explain that probability scales linearly with number, so this equation also applies for n as a probability

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

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

Response to voltage change How does the conductance change as we step the membrane potential? Time after start of pulse (ms) time 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!)

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

Measuring the parameters Hodgkin and Huxley summarized their data using algebraic expressions for the rate functions Remember that… OMG! What’s the point all of this?

Measuring the parameters Because once we have expressions for , we can integrate the differential equation for : Time after start of pulse (ms) Conductance To get the potassium conductance as a function of V and t:

Overview of the big picture 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: Start with at time step t Compute Integrate one time step to get Compute K current: Compute total membrane current: Compute Integrate to get at next time step

Transient conductances (sodium) Na and K conductances differ an important way: persistence activation activation inactivation Specific conductance (full scale =20 mS/cm2) Specific conductance (full scale =20 mS/cm2) Time after start of pulse (ms) Time after start of pulse (ms)

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

Sodium channel activation 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 τm as a function of voltage. milliseconds Membrane potential (mV) Time constant (ms) Note the similarity to K-channel activation… but 10x faster.

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

Sodium channel inactivation Dynamics of inactivation are captured by a new gating variable ‘h’. Specific conductance (full scale =20 mS/cm2) inactivation deinactivation time

Sodium channel inactivation How do we measure inactivation and recovery from inactivation? Hold Vm at different values Let the Na channels inactivate Then measure the Na current! Membrane potential (mV) holding potential measurement step milliseconds Membrane potential (mV) Time constant (ms) Replace with measurement of h-inf

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

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

Putting it all together! Start with initial contition at time step t0 Compute: Total membrane current Compute

Putting it all together!

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. Vc + Vout - Note that the series resistors form a voltage divider Ri Vm Ie Rm Vm Vout Ri Rm

Voltage Clamp Ie Rm + - Vc Vout Vm Ri Solve for Vm: Thus, for large G:

Note the similarity to K-channel activation… Potassium Sodium Membrane potential (mV) Membrane potential (mV) Membrane potential (mV) Time constant (ms) milliseconds Membrane potential (mV) Time constant (ms)