Biological Modeling of Neural Networks Week 2 – Biophysical modeling: The Hodgkin-Huxley model Wulfram Gerstner EPFL, Lausanne, Switzerland 2.1 Biophysics of neurons - Overview 2.2 Reversal potential - Nernst equation 2.3 Hodgin-Huxley Model 2.4 Threshold in the Hodgkin-Huxley Model - where is the firing threshold? 2.5. Detailed biophysical models - the zoo of ion channels Week 2 – part 1: Biophysics of neurons

2.1 Biophysics of neurons - Overview 2.2 Reversal potential - Nernst equation 2.3 Hodgin-Huxley Model 2.4 Threshold in the Hodgkin-Huxley Model - where is the firing threshold? 2.5. Detailed biophysical models - the zoo of ion channels Week 2 – part 1: Biophysics of neurons

Neuronal Dynamics – 2.1. Introduction visual cortex motor cortex frontal cortex to motor output

Neuronal Dynamics – 2.1. Introduction motor cortex frontal cortex to motor output neurons 3 km wires 1mm

Neuronal Dynamics – 2.1 Introduction neurons 3 km wires 1mm Signal: action potential (spike) action potential Ramon y Cajal How is a spike generated?

Review of week 1: Integrate-and-Fire models Spike emission -spikes are events -triggered at threshold -spike/reset/refractoriness Postsynaptic potential synapse t

Neuronal Dynamics – week 2: Biophysics of neurons Signal: action potential (spike) action potential Ca 2+ Na + K+K+ -70mV Ions/proteins Cell surrounded by membrane Membrane contains - ion channels - ion pumps

Neuronal Dynamics – week 2: Biophysics of neurons Ca 2+ Na + K+K+ -70mV Ions/proteins Cell surrounded by membrane Membrane contains - ion channels - ion pumps Resting potential -70mV  how does it arise? Ions flow through channel  in which direction? Na + Neuron emits action potentials  why?

Neuronal Dynamics – Biophysics of neurons Na + Resting potential -70mV  how does it arise? Ions flow through channel  in which direction? Neuron emits action potentials  why?  Hodgkin-Huxley model Hodgkin&Huxley (1952) Nobel Prize 1963

Neuronal Dynamics – Biophysics of neurons Na +  Hodgkin-Huxley model Hodgkin&Huxley (1952) Nobel Prize 1963

Neuronal Dynamics – Quiz In a natural situation, the electrical potential inside a neuron is [ ] the same as outside [ ] is different by microvolt [ ] is different by millivolt If a channel is open, ions can [ ] flow from the surround into the cell [ ] flow from inside the cell into the surrounding liquid Neurons and cells [ ] Neurons are special cells because they are surrounded by a membrane [ ] Neurons are just like other cells surrounded by a membrane [ ] Neurons are not cells Ion channels are [ ] located in the cell membrane [ ] special proteins [ ] can switch from open to closed Multiple answers possible!

Week 2 – Biophysical modeling: The Hodgkin-Huxley model Wulfram Gerstner EPFL, Lausanne, Switzerland 2.1 Biophysics of neurons - Overview 2.2 Reversal potential - Nernst equation 2.3 Hodgin-Huxley Model 2.4 Threshold in the Hodgkin-Huxley Model - where is the firing threshold? 2.5. Detailed biophysical models - the zoo of ion channels Week 2 – part 2: Reversal potential and Nernst equation Biological Modeling of Neural Networks

Neuronal Dynamics – 2.2. Resting potential Ca 2+ Na + K+K+ -70mV Ions/proteins Cell surrounded by membrane Membrane contains - ion channels - ion pumps Resting potential -70mV  how does it arise? Ions flow through channel  in which direction? Na + Neuron emits action potentials  why?

Neuronal Dynamics – Resting potential Na + Resting potential -70mV  how does it arise? Ions flow through channel  in which direction? Neuron emits action potentials  why?  Hodgkin-Huxley model Hodgkin&Huxley (1952) Nobel Prize 1963

100 mV 0 inside outside Ka Na Ion channels Ion pump density Mathetical derivation Ion pump  Concentration difference Neuronal Dynamics – Reversal potential E K

100 mV 0 inside outside Ka Na Ion channels Ion pump Neuronal Dynamics – Nernst equation K

100 mV 0 inside outside Ka Na Ion channels Ion pump Reversal potential Neuronal Dynamics – Nernst equation Concentration difference  voltage difference K

inside outside Ka Na Ion channels Ion pump Reversal potential Neuronal Dynamics – Reversal potential Ion pump  Concentration difference Concentration difference  voltage difference Nernst equation K

Reversal potential Exercise 1.1– Reversal potential of ion channels Calculate the reversal potential for Sodium Postassium Calcium given the concentrations What happens if you change the temperature T from 37 to 18.5 degree? Next Lecture 9:45

Neuronal Dynamics: Computational Neuroscience of Single Neurons Week 2 – Biophysical modeling: The Hodgkin-Huxley model Wulfram Gerstner EPFL, Lausanne, Switzerland 2.1 Biophysics of neurons - Overview 2.2 Reversal potential - Nernst equation 2.3 Hodgkin-Huxley Model 2.4 Threshold in the Hodgkin-Huxley Model - where is the firing threshold? 2.5. Detailed biophysical models - the zoo of ion channels Week 2 – part 3 : Hodgkin-Huxley Model

Neuronal Dynamics – Hodgkin-Huxley Model Na +  Hodgkin-Huxley model Hodgkin&Huxley (1952) Nobel Prize 1963 giant axon of squid

inside outside Ka Na Ion channels Ion pump CRlRl RKRK R Na I Mathematical derivation Hodgkin and Huxley, 1952 Neuronal Dynamics – 2.3. Hodgkin-Huxley Model

CRlRl RKRK R Na I Neuronal Dynamics – 2.3. Hodgkin-Huxley Model

100 mV 0 stimulus inside outside Ka Na Ion channels Ion pump Cglgl gKgK g Na I u u n 0 (u) Hodgkin and Huxley, 1952 Neuronal Dynamics – 2.3. Hodgkin-Huxley Model

u u r 0 (u) Neuronal Dynamics – 2.3. Ion channel

u u r 0 (u) Exercise 2 and 1.2 NOW!! - Ion channel Next lecture at: 10H40

Week 2 – Biophysical modeling: The Hodgkin-Huxley model Wulfram Gerstner EPFL, Lausanne, Switzerland 2.1 Biophysics of neurons - Overview 2.2 Reversal potential - Nernst equation 2.3 Hodgin-Huxley Model 2.4 Threshold in the Hodgkin-Huxley Model - where is the firing threshold? 2.5. Detailed biophysical models - the zoo of ion channels Week 2 – part 4: Threshold in the Hodgkin-Huxley Model Biological Modeling of Neural Networks

Neuronal Dynamics – 2.4. Threshold in HH model inside outside Ka Na Ion channels Ion pump

inside outside Ka Na Ion channels Ion pump u u h 0 (u) m 0 (u) Cglgl gKgK g Na I Where is the threshold for firing? Neuronal Dynamics – 2.4. Threshold in HH model

Constant current input I0I0 Cglgl gKgK g Na I Threshold? for repetitive firing (current threshold) Neuronal Dynamics – 2.4. Threshold in HH model

inside outside Ka Na Ion channels Ion pump pulse input I(t) Threshold? - AP if amplitude 7.0 units - No AP if amplitude 6.9 units (pulse with 1ms duration) (and pulse with 0.5 ms duration?) Neuronal Dynamics – 2.4. Threshold in HH model

Stim. u u h 0 (u) m 0 (u) pulse input I(t) Mathematical explanation Neuronal Dynamics – 2.4. Threshold in HH model

Stim. u u h 0 (u) m 0 (u) pulse input I(t) Neuronal Dynamics – 2.4. Threshold in HH model Where is the threshold? Blackboard!

Stim. u u h 0 (u) m 0 (u) pulse input I(t) Neuronal Dynamics – 2.4. Threshold in HH model Why start the explanation with m and not h? What about n? Where is the threshold?

Neuronal Dynamics – 2.4. Threshold in HH model

There is no strict threshold: Coupled differential equations ‘Effective’ threshold in simulations? First conclusion:

020ms Strong stimulus Action potential 100 mV 0 strong stimuli refractoriness Strong stimulus 100 mV 0 Where is the firing threshold? Refractoriness! Harder to elicit a second spike Neuronal Dynamics – 2.4. Refractoriness in HH model

I(t) 100 mV 0 Stimulation with time-dependent input current Neuronal Dynamics – 2.4. Simulations of the HH model

I(t) mV mV 0 Subthreshold response Spike Neuronal Dynamics – 2.4. Simulations of the HH model

Step current input I2I2 I Neuronal Dynamics – 2.4. Threshold in HH model

step input I2I2 I Where is the firing threshold? pulse input I(t) ramp input There is no threshold - no current threshold - no voltage threshold ‘effective’ threshold - depends on typical input Neuronal Dynamics – 2.4. Threshold in HH model

Response at firing threshold? ramp input/ constant input I0I0 Type I type II I0I0 I0I0 f f f-I curve Neuronal Dynamics – 2.4. Type I and Type II Hodgkin-Huxley model with standard parameters (giant axon of squid) Hodgkin-Huxley model with other parameters (e.g. for cortical pyramidal Neuron )

Neuronal Dynamics – 2.4. Hodgkin-Huxley model -4 differential equations -no explicit threshold -effective threshold depends on stimulus -BUT: voltage threshold good approximation Giant axon of the squid  cortical neurons -Change of parameters -More ion channels -Same framework

Exercise – Hodgkin-Huxley – ion channel dynamics stimulus inside outside Ka Na Ion channels Ion pump CgKgK I u u n 0 (u) Determine ion channel dynamics adapted from Hodgkin&Huxley 1952 apply voltage step voltage step u2u2 u Next Lecture at: 11.38

Week 2 – Biophysical modeling: The Hodgkin-Huxley model Wulfram Gerstner EPFL, Lausanne, Switzerland 2.1 Biophysics of neurons - Overview 2.2 Reversal potential - Nernst equation 2.3 Hodgkin-Huxley Model 2.4 Threshold in the Hodgkin-Huxley Model - where is the firing threshold? 2.5. Detailed biophysical models - the zoo of ion channels Week 2 – part 5: Detailed Biophysical Models Biological Modeling of Neural Networks

inside outside Ka Na Ion channels Ion pump Neuronal Dynamics – 2.5 Biophysical models  

Na+ channel from rat heart (Patlak and Ortiz 1985) A traces from a patch containing several channels. Bottom: average gives current time course. B. Opening times of single channel events Steps: Different number of channels Ca 2+ Na + K+K+ Ions/proteins Neuronal Dynamics – 2.5 Ion channels

inside outside Ka Na Ion channels Ion pump Neuronal Dynamics – 2.5 Biophysical models   There are about 200 identified ion channels How can we know which ones are present in a given neuron?

    Kv  1 Kv  2Kv  3 Kv  Voltage Activated K + Channels + + Kv1.1 Kv1.2Kv1.3Kv1.4Kv1.5 Kv1.6 Kv2.2Kv2.1 Kv3.2 Kv3.1 Kv3.3 Kv3.4 KChIP1 KChIP2KChIP3 KChIP Kv1Kv3Kv2Kv4Kv5Kv6Kv8Kv9 Kv Kv4.3 Kv4.1 Kv4.2 Ion Channels investigated in the study of CGRP CB PV CR NPY VIP SOM CCK pENK Dyn SP CRH schematic mRNA Expression profile Cglgl g Kv1 g Na I g Kv3 Toledo-Rodriguez, …, Markram (2004)

Cglgl g Kv1 g Na I g Kv3 Model of a hypothetical neuron stimulus inside outside Ka Na Ion channels Ion pump Erisir et al, 1999 Hodgkin and Huxley, 1952 How many parameters per channel?

Cglgl g Kv1 g Na I g Kv3 Model of a hypothetical neuron I(t) Spike Subthreshold Detailed model, based on ion channels

Cglgl g Kv1 g Na I g Kv3 Model of a hypothetical neuron (type I) Biophysical model, based on ion channels Current pulse constant current - Delayed AP initiation - Smooth f-I curve type I neuron

Neuronal Dynamics – 2.5 Adaptation Functional roles of channels? - Example: adaptation I u

Neuronal Dynamics – 2.5 Adaptation: I M -current M current: - Potassium current - Kv7 subunits - slow time constant I M current is one of many potential sources of adaptation Yamada et al., 1989

Neuronal Dynamics – 2.5 Adaptation – I NaP current current: - persistent sodium current - fast activation time constant - slow inactivation ( ~ 1s) I NaP current - increases firing threshold - source of adaptation Aracri et al., 2006

inside outside Ka Na Ion channels Ion pump Neuronal Dynamics – 2.5 Biophysical models  Hodgkin-Huxley model provides flexible framework Hodgkin&Huxley (1952) Nobel Prize 1963

Exercise 4 – Hodgkin-Huxley model – gating dynamics A) Often the gating dynamics is formulated as B) Assume a form How are a and b related to and in the equations C ) What is the time constant ? Calculate

Now Computer Exercises: Play with Hodgkin-Huxley model

Neuronal Dynamics – References and Suggested Reading Reading: W. Gerstner, W.M. Kistler, R. Naud and L. Paninski, Neuronal Dynamics: from single neurons to networks and models of cognition. Chapter 2: The Hodgkin-Huxley Model, Cambridge Univ. Press, 2014 OR W. Gerstner and W. M. Kistler, Spiking Neuron Models, Chapter 2, Cambridge, Hodgkin, A. L. and Huxley, A. F. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol, 117(4): Ranjan, R.,et al. (2011). Channelpedia: an integrative and interactive database for ion channels. Front Neuroinform, 5:36. -Toledo-Rodriguez, M., Blumenfeld, B., Wu, C., Luo, J., Attali, B., Goodman, P., and Markram, H. (2004). Correlation maps allow neuronal electrical properties to be predicted from single-cell gene expression profiles in rat neocortex. Cerebral Cortex, 14: Yamada, W. M., Koch, C., and Adams, P. R. (1989). Multiple channels and calcium dynamics. In Koch, C. and Segev, I., editors, Methods in neuronal modeling, MIT Press. - Aracri, P., et al. (2006). Layer-specic properties of the persistent sodium current in sensorimotor cortex. Journal of Neurophysiol., 95(6):