1. Romain Brette Ecole Normale Supérieure, Paris romain.brette@ens.fr What is the most realistic single-compartment neuron model?

2. The simplicity-realism trade-off Izhikevich (2004). Which model to use for cortical neurons? IEEE Neural networks 15(5)

3. What is a « realistic »model? - lots of features named after the real object - it flies sodium channels potassium channels spike threshold - abstract

4. What is a « realistic »model? - lots of features named after the real object - it flies sodium channels potassium channels spike threshold - abstract

5. The Hodgkin-Huxley model Hodgkin & Huxley (Nobel prize 1963)

6. The Hodgkin-Huxley model Hodgkin & Huxley (Nobel prize 1963) Model of the space clamped giant squid axon

7. In what sense is it « realistic »? several equations and variables although phenomenological! but you can make predictions about Vm, and about Na and K currents

8. In what sense is it « realistic »? several equations and variables although phenomenological! you can predict voltage-clamp measurements

9. The empirical content of a model Karl Popper:“I define the empirical content of a statement p as the class of its potential falsifiers” (The logic of scientific discovery, 1935). In other words: what a model can potentially predict Integrate-and-fire modelQuadratic model Hodgkin-Huxley model Vm (subthreshold) spikes Vm I Na IKIK

10. Empirical content of neuron models multicompartmental Hodgkin-Huxley model single-compartment Hodgkin-Huxley model integrate-and-fire model exponential integrate-and-fire model quadratic integrate-and-fire model empirical content

11. Correctness of a model Integrate-and-fire modelQuadratic model Hodgkin-Huxley model Vm (subthreshold) spikes Vm I Na IKIK Can make more predictions, but are predictions correct?

12. Membrane potential (Vm) Naundorf, Wolf, Volgushev (Nature 2006) 1) Spikes have sharp onsets in recordings, unlike in single compartment Hodgkin-Huxley models Badel et al. (J Neurophysiol 2008) V 2) I-V relationship at soma is very sharp, unlike in single compartment Hodgkin-Huxley models

13. Sodium current (INa) Milescu, Bean, Smith (J Neurosci 2010) -45 mV -25 mV I Na (nA) Somatic voltage-clamp Cortical pyramidal neuron Inferior olivary neuronMotoneuron Voltage-clamp recordings are not as predicted by the single-compartment HH model mV soma total

14. Correctness of the HH model Integrate-and-fire model Hodgkin-Huxley model Vm (subthreshold) spikes Vm I Na IKIK The additional empirical content is not correct!

15. Correctness of the HH model Integrate-and-fire model Hodgkin-Huxley model Vm (subthreshold) spikes Vm I Na IKIK The additional empirical content is not correct! How about spikes?

16. Predicting spikes The INCF competition Gerstner & Naud (Science 2009)

17. Results for IF models cortical neuron MAT2 (86%) AIF (84%) Izhikevich (62%) Rossant et al. (Frontiers Neurosci 2011)

18. And the Hodgkin-Huxley model? We did not manage to fit it However, we did fit an approximation of it, the adaptive exponential IF model Fourcaud-Trocme et al. (J Neurosci 2003) Brette & Gerstner (J Neurophysiol 2005) is the « slope factor » = Na activation slope k a Patch-clamp measurements: k a = 6 mV Best prediction of spikes:= 0 mV the best fit is an IF model!

19. Transfer function of cortical neurons Ilin et al. (J Neurosci 2013) 3-6 Hz Prediction of single-compartment HH models: cut-off frequency should be an order of magnitude lower

20. Transfer function of cortical neurons Ilin et al. (J Neurosci 2013) 3-6 Hz Fast response to current changes, as in IF models! Brunel et al. (PRL 2001)

21. Correctness of the HH model Integrate-and-fire model Hodgkin-Huxley model Vm (subthreshold) spikes Vm I Na IKIK

22. Correctness of the HH model Integrate-and-fire model Hodgkin-Huxley model Vm (subthreshold) spikes Vm I Na IKIK Spikes are also not well predicted by the single-compartment HH model!

23. Correctness of neuron models single-compartment Hodgkin-Huxley model integrate-and-fire model exponential integrate-and-fire model quadratic integrate-and-fire model correctness But how is this possible ???

24. Spikes are initiated in the axon Stuart, Schiller, Sakmann (J Physiol 1997)

25. Spikes are initiated in the axon Stuart, Schiller, Sakmann (J Physiol 1997) Debanne et al. (Phys Rev 2011)

26. Compartmentalization Most of the time, Vm is the same in soma and AIS At spike initiation, the two compartments are decoupled Kole, Letzkus, Stuart (Neuron 2007) AIS

27. Compartmentalization V a – V s = axonal current * R a VaVa VsVs Milescu, Bean, Smith (J Neurosci 2010) axonal current in voltage-clamp

28. Model of axonal initiation soma I VsVs VaVa R a.I The soma is a current sink (Ohm’s law) Brette, R. (2013). Sharpness of spike initiation in neurons explained by compartmentalization. PLoS Comp Biol (in press)

29. Model of axonal initiation soma I VsVs VaVa R a.I The soma is a current sink (Ohm’s law)

30. Model of axonal initiation soma I=f(V a ) VsVs VaVa R a.I Na activation

31. Model of axonal initiation soma I=f(V a ) I=(V a -V s )/Ra VsVs I (nA) I=f(V a ) I=(V a -V s )/Ra VsVs Lateral and Na currents must match Discrete opening of Na channels

32. Na channels near the soma soma I=f(V a ) I=(V a -V s )/Ra VsVs I (nA) I=f(V a ) I=(V a -V s )/Ra VsVs Continuous opening of Na channels

33. A view from the soma m Lateral current flows abruptly when a voltage threshold is exceeded Na channels open in an all-or-none fashion

34. A view from the soma m single compartment HH model with axonal initiation A fairly good phenomenological description: -below V t, no sodium current -when Vm reaches V t : all channels open (spike) VtVt a.k.a. the integrate-and-fire model !

35. Value of spike threshold soma I=f(V a ) I=(V a -V s )/Ra VsVs Lateral and Na currents must match f(V a ) = (V a -V s )/Ra Fixed point equation Na activation Spike threshold = bifurcation point (= V s when solution jumps) I (nA) The threshold equation V 1/2 kaka

36. Conclusion looks good but doesn’t fly!it flies! Single-compartment Hodgkin-Huxley modelIntegrate-and-fire model Brette, R. (2013). Sharpness of spike initiation in neurons explained by compartmentalization. PLoS Comp Biol (in press)