Basic neurophysics Gregor Schöner Institut für Neuroinformatik sources (except where cited otherwise)

Peter Dayan and Larry F Abbot: Theoretical , MIT Press, Cambridge MA, 2001

sections 1.1, 1.2, 1.4, 2.3 , Werner M. Kistler, Richard Naud and Liam Paninski: Neuronal Dynamics: From single to networks and models of cognition. Cambridge University Press, 2014

section 2 http://neuronaldynamics.epfl.ch/index.html Neuronal Dynamics – 1.1. Neurons and Synapses/Overview the motor cortex How do we recognize? frontal Models of cogntion visual cortex Weeks 10-14 cortex

to motor output Neuronal Dynamics – 1.1.neurons Neurons and Synapses/Overview

motor 1mm 10 000 neurons 3 km wires cortex frontal cortex

to motor output

~10^11 with 10000 synapses each neurons neurons as input-output units

inputs from dendrites spike formation at soma output at axon two functional components membranes: dendrites, soma, axons synapses membrane

source http://www.columbia.edu/cu/psychology/courses/1010/mangels/neuro/neurosignaling/neurosignaling.html] membrane

membrane=double lipid layer that is an electrical insulator is electrically charged: more negative potential inside than outside cell based on ions K+, Na+, and Cl-

source http://www.columbia.edu/cu/psychology/courses/1010/mangels/neuro/neurosignaling/neurosignaling.html] membrane higher concentration of K+ inside cell lower concentration of Na+ inside cell membrane less permeable to Na+ than to K+ => Na+ gradient is steeper than the K+ gradient => more positive outside cell => negative potential

source http://www.columbia.edu/cu/psychology/courses/1010/mangels/neuro/neurosignaling/neurosignaling.html] membrane

gradient comes from ion pumps: protein channels in membrane that transport Na+ out of cell, K+ into cell, establishing gradient this is where energy is consumed (a lot): ATP used to pump ions

source http://www.columbia.edu/cu/psychology/courses/1010/mangels/neuro/neurosignaling/neurosignaling.html] membrane giant squid axon… used to establish basic of membrane dynamics

voltage-clampFigure 3.1, [Current flow across a squid...]. - Neuroscience - NCBI Bookshelf 26 Nov 2015, 10:44

Figure 3.1 [Source: Neuroscience. 2nd edition. Purves D, Augustine GJ, Fitzpatrick D, et al., editors. Current flow across a squid axon membrane during a voltage clamp experiment. (A) A 65 mV hyperpolarization of the membrane potential produces only Sunderlanda very brief capacitive (MA): current.Sinauer (B) Associates; A 65 mV depolarization 2001.] of the membrane potential also produces a brief capacitive current which is followed by a longer-lasting but transient phase of inward current and a delayed but sustained outward current. (After Hodgkin et al., 1952.)

From: Ionic Currents Across Nerve Cell Membranes Neuroscience. 2nd edition. Purves D, Augustine GJ, Fitzpatrick D, et al., editors. Sunderland (MA): Sinauer Associates; 2001.

Copyright © 2001, Sinauer Associates, Inc. NCBI Bookshelf. A service of the National Library of , National Institutes of Health.

http://www.ncbi.nlm.nih.gov/books/NBK10879/figure/A178/?report=objectonly Page 1 of 1 synapses

at a synapse, the membranes of two neurons comes very close => this is where transmission across neurons takes place two types of synapses

electrical: currents across the membrane directly from one cell to another through “gap junctions”

very fast, but not flexible. exists in the peripheral … but not very common chemical: the common one that is much more flexible…

[Source: Neuroscience. 2nd edition. Purves D, Augustine GJ, Fitzpatrick D, et al., editors. Sunderland (MA): Sinauer Associates; 2001.] two types of synapses

chemical: the more common one pre-synaptic cell releases in response to an that arrives through the axon post-synaptic potential induced by action of neurostransmitters on receptors

[Source: Neuroscience. 2nd edition. Purves D, Augustine GJ, Fitzpatrick D, et al., editors. Sunderland (MA): Sinauer Associates; 2001.] two types of synapses

chemical synapse slower transmission… 1 to 2 ms but more flexible: tuned by changes in receptors

[Source: Neuroscience. 2nd edition. Purves D, Augustine GJ, Fitzpatrick D, et al., editors. Sunderland (MA): Sinauer Associates; 2001.] post-synapticFigure 7.7, [Summation of postsynapticpotentials potentials. (A)...]. - Neuroscience - NCBI Bookshelf 26 Nov 2015, 13:36

depending on the receptor type, synaptic transmission induces post- synaptic potentials of different forms and sign that travel to the soma, where a spiking decision is made [Source: Neuroscience. 2nd edition. Purves D, Augustine GJ, Fitzpatrick D, et al., editors. Figure 7.7 Summation of postsynaptic potentials. (A) A microelectrode records the postsynaptic potentials produced by the activity of two excitatory synapses (E1 and E2) and an inhibitory synapse (I). (B) Electrical responses to synaptic activation. Stimulating either excitatory synapse (E1 or E2) produces a subthreshold EPSP, whereas stimulating both synapses at the same time (E1 + E2) produces a suprathreshold EPSP that evokes a postsynaptic action potential (shown in blue). Activation of the inhibitory synapse alone (I) results in a hyperpolarizing IPSP. Summing this IPSP (dashed red line) with the EPSP (dashed yellow line) produced by one excitatory synapse (E1 + I) reduces the amplitude of the EPSP (orange line), while summing it with the suprathreshold EPSP produced by activating synapses E1 and E2 keeps the postsynaptic neuron below threshold, so that no action potential is evoked.

From: Summation of Synaptic Potentials Neuroscience. 2nd edition. Purves D, Augustine GJ, Fitzpatrick D, et al., editors. Sunderland (MA): Sinauer Associates; 2001.

Copyright © 2001, Sinauer Associates, Inc. NCBI Bookshelf. A service of the National Library of Medicine, National Institutes of Health.

http://www.ncbi.nlm.nih.gov/books/NBK11104/figure/A480/?report=objectonly Page 1 of 1 Neuronal Dynamics – 1.2. The passivespiking membrane mechanism

electrode Spike emission

u

synapse t potential Integrate-and-fire model spiking mechanism all or Neuronalnone nature Dynamics of spikes – 1.2. The passive membrane spike generation is electrode coincidence detection Spike emission

u overlap of incoming post-synaptic potentials that have propagated to soma within about 10 ms required synapse t to sum… potential typical in cortex: 10 inputs needed, 10000 potential inputs… Integrate-and-fire model neuron as a “switch” Hodgkin-Huxley Neuronal Dynamics – 2. 2. Nernst equation inside 100 KaK

mV relationship Na potential-ionicNeuronal Dynamics – 2. 2. Nernst equation outside 0 Ion channels Ion pump inside concentration100 KaK

mV Na kT nu()1 0 outside Ion channels Ion pump uu12 u ln nu() kT 2 nu()1 q uu12 u ln q nu()2 Reversal potential Reversal potential Concentration difference voltage differenceConcentration difference voltage difference 2.2 Hodgkin-Huxley Model | Neuronal Dynamics online book 26 Nov 2015, 10:50

2 The Hodgkin-Huxley Model 2.1 Equilibrium potential 2.3 The Zoo of Ion Channels

Toggle navigation Neuronal Dynamics

About Online book Video Lectures Teaching Material

2.2 Hodgkin-Huxley Model | Neuronal Dynamics online book 26 Nov 2015, 10:50 2.2 Hodgkin-Huxley Model there are separate batteries for sodium, potassium, and the unspecific third channel, with Hodgkin and Huxley (222) performed experiments on the giant axon of the squid and found battery voltages ENa, EKthree and different EL , respectively. types of ion current, viz., sodium, potassium, and a leak current that consists mainly of Cl− ions. Specific voltage-dependent ion channels, one for sodium and another one Let us now translate the abovefor potassium, schema control of an the electrical flow of those circuit ions through into mathematical the cell membrane. equations. The leak Thecurrent takes 2.2 Hodgkin-Huxley Model | Neuronal Dynamics online book 26 Nov 2015, 10:50 conservation of electric chargecare of otheron a channel piece oftypes membrane which are notimplies described that explicitly. the applied current I (t) may be split in a capacitivechannels current may beIC characterized which charges by their the resistancecapacitor or, C equivalently, and further by components their conductance. The which pass through the ion channels. Thus 2.2.1 Definition of the model Ik leakage channel is described by a voltage-independent conductance gL = 1/R . Since u is the Hodgkin-Huxleytotal voltage across the cell membrane and EL the voltage of the battery, the voltage at the leak resistor in Fig. 2.2 is u − EL . Using Ohm’s law, we get a leak current IL = gL (u − EL) . I t I t I t The mathematics( ) of= theC other( ) +ion∑ channelsk ( )is analogous except that their conductance(2.3) is voltage and time dependent. If all kchannels are open, they transmit currents with a maximum conductance or , respectively. Normally, however, some of the channels are blocked. gNa gK The breakthrough of Hodgkin and Huxley was that they succeeded to measure how the where the sum runs overeffective all ion resistance channels. of Ina channel the standard changes Hodgkin-Huxley as a function of time model and voltage. there are Moreover, they dynamic modelonly three types of of channel:proposed a sodium a mathematical channel description with index of Na,their aobservations. potassium channelSpecifically, with they index introduced K and an unspecific leakageadditional channel ’gating’ with variables resistance m, n Rand; cf. h to Fig. model 2.2 the. From probability the definition that a channel of a is open at a potential capacitychange C = qand/u wheregiven q momentis a charge in time. and The u the combined voltage action across of mthe and capacitor, h controls we the find Na+ the channels while the K + Fig. 2.2: Schematic diagram for the Hodgkin-Huxley model. charging current IC = C gatesdu/ aredt .controlled Hence from by n (.2.3 For) example, the effective conductance of sodium channels is modeled as 3 , where describes the activation (opening) of the channel three ion currents 1/RNa = gNa m h m and its inactivation (blocking). The conductance of potassium is 4 , where 2.2 Hodgkin-Huxley Model | Neuronal DynamicsTheh Hodgkin-Huxley online book model can be understood with the help of Fig. 2.21/.R TheK = semipermeablegK n 26 Nov 2015,n 10:50 describes thed activationu of the channel. cell membraneC separates the interiort of theI cell from the extracellular liquid and acts(2.4) as a capacitor. If an input= − current Ik ( )is +injected(t) into. the cell, it may add further charge on the which come from In summary, Hodgkint and∑ HuxleyI (t) formulated the three ion currents on the right-hand-side of capacitor, ord leak throughk the channels in the cell membrane. Each channel type is represented three ion channels Eq.in (Fig.2.4) 2.2 as by a resistor. The unspecific channel has a leak resistance R, the sodium channel a Example:resistance Time RNa Constants, and the potassium Transition channel Rates, a resistance and RChannelK . The diagonal Kinetics arrow across the In biological terms, u is diagramthe voltage of the across resistor theindicates membrane that the valueand of theI resistance is the sum is not of fixed, the butionic changes depending on whether the ion3 channel is open ∑or closed.k k4 Because of active ion transport currents which pass through the cellI membrane.k = g m h (u − E ) + g n (u − E ) + gL (u − EL) . (2.5) As an alternativethrough to the the∑ formulation cell membrane,Na of channelthe ion concentration gatingNa in Eq. Kinside (2.6 the), thecell activationisK different fromand that in the k phenomenologicalinactivation dynamicsextracellular of each liquid. channel The Nernst type potential can also generated be described by the differencein terms ofin ionvoltage- concentration is dependent transitionrepresented rates by α aand battery β, in Fig. 2.2. Since the Nernst potential is different for each ion type, dynamics of the ion The parameters E , E , and E are the reversal potentials. A http://neuronaldynamics.epfl.ch/online/Ch2.S2.htmlNa K B L Page 1 of 13 m˙ = αm u (1 − m) − βm u m ion channels ( ) ( ) 2 x Ex [mV] gx [mS / cm ] n˙ = αn (u) (1 − n) − βn (u) n (2.9) Na 55 40 h˙ = αh (u) (K1 − h)-77− βh (u) h.35 L -65 0.3 The two formulations Eqs. (2.6) and (2.2.2) are equivalent. The asymptotic value x0 (u) and the time constant τx (u) are given by the transformation −1 −1 x αx u [ ] βx u −1 [ ] x0 (u) = αx (u) / [αx (u) +( β/x mV(u) ])andms τx (u) = [αx (u) + βx((u/)]mV. The) ms various functionsn α and0.02 β, (givenu − 25 in) Table/ [1 2.1− e, −are(u− empirical25) / 9] −functions0.002 ( uof− u25 that) / produce[1 − e(u−25) / 9] the curves in Figurem 2.30.182. (u + 35) / [1 − e−(u+35) / 9] −0.124 (u + 35) / [1 − e(u+35) / 9] h e−(v+90)/12 e(v+62)/6 e(v+90)/12 Equations (2.2.2) are typical equations0. 25 used in chemistry to describe0. the25 stochastic/ dynamicsFig. of2.3: an Theactivation Hodgkin-HuxleyTable process 2.1: Parameters with model. rate forconstants Athe. TheHodgkin-Huxley α equilibrium and β. We equations may interpret fitted onthis processfunctions as a molecular for the pyramidal switchthree variables between neurons twom of, thenstates, hcortex. in with the The voltage-dependentHodgkin-Huxley parameters for n andtransition m were rates. fitted For example,model. Bthe. The activation voltageby Zach variable dependent Mainen n (can323 time )be on interpreted constant.experiments The as reported the resting probability by Huguenard of finding et al. a(233) singlepotential potassium is channelat u and= open.− the65 parameters mThereforeV (arrow) for in h aand bypatch Richard parameters with NaudK channels, areon thethose experiments approximately reported in given in Table 2.1Hamill. et al. (205). Voltage is measured in mV and the membrane k ( n) K channels are expected to be closed. We may interpret αn u t as the ≈ 1 − capacity is C = 1μ 2 . ( ) Δ probability that in a short time interval ΔtF/cm one of the momentarily closed channels switches to theThe open three state. gating variables m, n, and h evolve according to differential equations of the form As mentioned above, the Hodgkin-Huxley model describes three types of channel. All http://neuronaldynamics.epfl.ch/online/Ch2.S2.html Page 3 of 13

http://neuronaldynamics.epfl.ch/online/Ch2.S2.html Page 2 of 13

2.2.3 Dynamics

A

B

http://neuronaldynamics.epfl.ch/online/Ch2.S2.html Page 7 of 13 Hodgkin-Huxley

based on data from squid-axon…

2.2 Hodgkin-Huxley Model | Neuronal Dynamics online book 26 Nov 2015, 10:50

Fig. 2.4: Original data and fit of Hodgkin and Huxley (1952). The measured time course of the potassium conductance (circles) after application of a voltage step of 25mV (left, marked A) and after return to resting potential (right, B). The fit (solid line) is based on Eq. (2.8). Adapted from (222).

Example: Activation and De-inactivation

The variable m is called an activation variable. To understand this terminology, we note from Fig. 2.3 that the value of m0 (u) at the neuronal resting potential of u=-65mV is 3 close to zero. Therefore, at rest, the sodium current INa = gNa m h (u − ENa) through the channel vanishes. In other words, the sodium channel is closed.

When the membrane potential increases significantly above the resting potential, the gating variable m increases to its new value m0 (u). As long as h does not change, the sodium current increases and the gate opens. Therefore the variable m ’activates’ the channel. If, after a return of the voltage to rest, m decays back to zero, it is said to be ‘de-activating’.

The terminology of the ‘inactivation’ variable h is analogous. At rest, h has a large positive value. If the voltage increases to a value above -40mV, h approaches a new value h0 (u) which is close to rest. Therefore the channel ‘inactivates’ (blocks) with a time constant that is given by τh (u). If the voltage returns to zero, h increases so that the channel undergoes ‘de-inactivation’. This sounds like a tricky vocabulary, but it turns out to be useful to distinguish between a deactivated channel (m close to zero and h close to one) and an inactivated channel (h close to zero).

2.2.2 Stochastic Channel Opening

The number of ion channels in a patch of membrane is finite and individual ion channels open and close stochastically. Thus, when an experimentalist records the current flowing through a small patch of membrane, he does not find a smooth and reliable evolution of the measured variable over time but rather a highly fluctuating current, which looks different at each

http://neuronaldynamics.epfl.ch/online/Ch2.S2.html Page 5 of 13 2.2 Hodgkin-Huxley Model | Neuronal Dynamics online book 26 Nov 2015, 10:50

Example: Time Constants, Transition Rates, and Channel Kinetics

As an alternative to the formulation of channel gating in Eq. (2.6), the activation and inactivation dynamics of each channel type can also be described in terms of voltage- dependent transition rates α and β,

m˙ = αm (u) (1 − m) − βm (u) m n˙ = αn (u) (1 − n) − βn (u) n (2.9) h˙ = αh (u) (1 − h) − βh (u) h.

The two formulations Eqs. (2.6) and (2.2.2) are equivalent. The asymptotic value x0 (u) and the time constant τx (u) are given by the transformation −1 x0 (u) = αx (u) / [αx (u) + βx (u) ]and τx (u) = [αx (u) + βx (u)] . The various functions α and β, given in Table 2.1, are empirical functions of u that produce the curves in Figure 2.3.

Equations (2.2.2) are typical equations used in chemistry to describe the stochastic dynamics of an activation process with rate constants α and β. We may interpret this process as a molecular switch between two states with voltage-dependent transition rates. For example, the activation variable n can be interpreted as the probability of finding a single potassium channel open. Therefore in a patch with K channels, approximately k ≈ (1 − n) K channels are expected to be closed. We may interpret αn (u) Δt as the probability that in a short time interval Δt one of the momentarily closed channels switches to the open state.

2.2.3 Dynamics

A Hodgkin-Huxley

2.2 Hodgkin-Huxley Model | Neuronal Dynamics online book 26 Nov 2015, 10:50 spikes B

C http://neuronaldynamics.epfl.ch/online/Ch2.S2.html Page 7 of 13

Fig. 2.6: A. Action potential. The Hodgkin-Huxley model is stimulated by a short, but strong, current pulse between t = 1 and t = 2 ms. The time course of the membrane potential u (t) for t > 2 ms shows the action potential (positive peak) followed by a relative refractory period where the potential is below the resting potential (dashed line). The right panel urest shows an expanded view of the action potential between t = 2 and t = 5 ms. B. The dynamics of gating variables m, h, n illustrate how the action potential is mediated by sodium and potassium channels. C. The sodium current which depends INa on the variables m and h has a sharp peak during the upswing of an action potential. The potassium current is controlled IK by the variable and starts with a delay compared to . n INa

In this subsection we study the dynamics of the Hodgkin-Huxley model for different types of input. Pulse input, constant input, step current input, and time-dependent input are considered in turn. These input scenarios have been chosen so as to provide an intuitive understanding of the dynamics of the Hodgkin-Huxley model.

The most important property of the Hodgkin-Huxley model is its ability to generate action potentials. In Fig. 2.6A an action potential has been initiated by a short current pulse of 1 ms duration applied at t = 1 ms. The spike has an amplitude of nearly 100mV and a width at half maximum of about 2.5ms. After the spike, the membrane potential falls below the resting potential and returns only slowly back to its resting value of -65mV.

Ion channel dynamics during spike generation

In order to understand the biophysics underlying the generation of an action potential we return to Fig. 2.3A. We find that and increase with whereas decreases. Thus, if m0 n0 u h0 some external input causes the membrane voltage to rise, the conductance of sodium channels http://neuronaldynamics.epfl.ch/online/Ch2.S2.html Page 8 of 13 2.2 Hodgkin-Huxley Model | Neuronal Dynamics online book 26 Nov 2015, 11:02

Example: Mean firing rates and gain function

The Hodgkin-Huxley equations (2.4)-(2.2.2) may also be studied for constant input I (t) = I0 for t > 0. (The input is zero for t ≤ 0). If the value I0 is larger than a 2 critical value Iθ ≈ 2.7μA/cm , we observe regular spiking; Fig. 2.7A. We may define a firing rate ν = 1/T where T is the inter-spike interval.

The firing rate as a function of the constant input I0 , often called the ’frequency-current’ relation or ‘f-I-plot’, defines the gain function plotted in Fig. 2.7B. With the parameters given in Table 2.1, the gain function exhibits a jump at Iθ . Gain functions with a discontinuity are called ’type II’.

If we shift the curve of the inactivation variable h to more positive voltages, and keep otherwise the same parameters, the modified Hodgkin-Huxley model exhibits a smooth gain function; see Section 2.3.2 and Fig. 2.11. Neuron models or, more generally, ’excitable membranes’ are called ’type I’ or ’class I’ if they have a continuous frequency- current relation. The distinction between excitability of type I and II can be traced back to Hodgkin (223). Hodgkin Huxley

the spiking mechanism is an instability => threshold effect

A B

Fig. 2.8: A. Spike train of the Hodgkin-Huxley model driven by a time dependent input current. The action potentials occur irregularly. The figure shows the voltage u as a function of time. B. Threshold effect. A short current pulse of 1ms is applied which leads to a excursion of the membrane potential of a few millivolt (dashed line). A slight increase of the strength of the current pulse leads to the generation of an action potential (solid line) with an amplitude of about 100mV above rest (out of bounds).

http://neuronaldynamics.epfl.ch/online/Ch2.S2.html Page 10 of 13 2.2 Hodgkin-Huxley Model | Neuronal Dynamics online book 26 Nov 2015, 11:02

increases due to increasing m. As a result, positive sodium ions flow into the cell and raise the membrane potential even further. If this positive feedback is large enough, an action potential is initiated. The explosive increase comes to a natural halt when the membrane potential approaches the reversal potential of the sodium current. HodgkinENa Huxley At high values of u the sodium conductance is slowly shut off due to the factor h. As indicated in Fig. 2.3B, the ‘time constant’ τh is always larger than τm . Thus the variable h which inactivates the channels reacts more slowly to the voltage increase than the variable m which opens the channel. On a similar slow time scale, the potassium (K+) current sets in Fig. 2.6C. spikeSince rate it is a currentreflects in outward input direction, current it lowers the potential. The overall effect of the sodium and potassium currents is a short action potential followed by a negative overshoot; cf. Fig. 2.6A. The negative overshoot, called hyperpolarizing spike-after potential, is due to the slow de-inactivation of the sodium channel, caused by the h-variable.

A B

C D

Fig. 2.7: A. Spike train of the Hodgkin-Huxley model (with the parameters used in this book) for constant input current . B. I0 Gain function. The mean firing rate ν is plotted as a function of . The gain function of the Hodgkin-Huxley model is of type I0 II, because it exhibits a jump. C. Same as A, but for the original parameters found by Hodgkin and Huxley to describe the ion currents in the giant axon of the squid. D. Gain function for the model in C.

http://neuronaldynamics.epfl.ch/online/Ch2.S2.html Page 9 of 13 2.2 Hodgkin-Huxley Model | Neuronal Dynamics online book 26 Nov 2015, 11:02

Example: Mean firing rates and gain function

The Hodgkin-Huxley equations (2.4)-(2.2.2) may also be studied for constant input I (t) = I0 for t > 0. (The input is zero for t ≤ 0). If the value I0 is larger than a 2 critical value Iθ ≈ 2.7μA/cm , we observe regular spiking; Fig. 2.7A. We may define a firing rate ν = 1/T where T is the inter-spike interval.

The firing rate as a function of the constant input I0 , often called the ’frequency-current’ relation or ‘f-I-plot’, defines the gain function plotted in Fig. 2.7B. With the parameters given in Table 2.1, the gain function exhibits a jump at Iθ . Gain functions with a discontinuity are called ’type II’.

If we shift the curve of the inactivation variable h to more positive voltages, and keep otherwise the same parameters, the modified Hodgkin-Huxley model exhibits a smooth gain function; see Section 2.3.2 and Fig. 2.11. Neuron models or, more generally, Hodgkin ’excitableHuxley membranes’ are called ’type I’ or ’class I’ if they have a continuous frequency- current relation. The distinction between excitability of type I and II can be traced back to Hodgkin (223). time varying inputs make time varying rate

A B

Fig. 2.8: A. Spike train of the Hodgkin-Huxley model driven by a time dependent input current. The action potentials occur irregularly. The figure shows the voltage u as a function of time. B. Threshold effect. A short current pulse of 1ms is applied which leads to a excursion of the membrane potential of a few millivolt (dashed line). A slight increase of the strength of the current pulse leads to the generation of an action potential (solid line) with an amplitude of about 100mV above rest (out of bounds).

http://neuronaldynamics.epfl.ch/online/Ch2.S2.html Page 10 of 13 Example:

stretch reflex

[Source: Neuroscience. 2nd edition. Purves D, Augustine GJ, Fitzpatrick D, et al., editors. Sunderland (MA): Sinauer Associates; 2001.]