The dynamics of bursting in neurons and networks. Professor Jonathan Rubin January 20, 2011

Transcription

1 The dynamics of bursting in neurons and networks Professor Jonathan Rubin January 20, 2011

2 Outline: Introduction to neurons Introduction to bursting Mathematical analysis of bursting in single cells Frontiers in research on bursting dynamics (21-53)

3 a cartoon neuron

4 4-100 µm in diameter, ~10-6 grams each; ~25,000 m 2 surface area over whole human brain (like 4 soccer fields)

5 patch-clamp recordings (CNRS, France)

6 action potential generation OUTSIDE cell membrane INSIDE

7 examples of bursting in neurons (Scholarpedia) multiple action potentials at relatively high frequency (100 Hz) long interburst intervals

8 why care about bursting? 1. mathematically interesting 2. plays a role in the brain (a) sleep (b) novelty detection

9 (c) central pattern generators (CPGs) crustacean STG Rabbeh and Nadim, J. Neurophysiol., 2007

10 Nat. Rev. Neurosci., 2005 (d) and pathology Raz et al., J. Neurosci., 2000

11 mathematical modeling/analysis: single-neuron bursting example: the respiratory pre- Bötzinger complex (prebötc) embedded in network isolated Butera, Rinzel, Smith, 1999

12 bursting in differential equation models voltage chaotic periodic time

13 Butera et al. model Butera, Rinzel, Smith, 1999

14 fast-slow decomposition for bursting analysis (Rinzel, 1985+)

15 fast/slow decomposition for the Butera et al. model fast slow inactivation of persistent sodium (h) deinactivation of persistent sodium (h) slow in this example, there is a stable fixed point corresponding to quiescence

16 conditional square-wave bursting in the Butera et al. model A: quiescent B,C: bursting D: tonic spiking g tonic-e

17 rigorous framework Terman,

18 full classification? Izhikevich, 2000 (pg ) slow fast

19 case NOT closed e.g., the Bursting book, 2005

20 1) noise Kuske and Baer, Bull. Math. Biol., 2002 Su, R. and Terman, Nonlinearity, 2004 Pedersen and Sorensen, SIAP, 2007

21 2) multiple bifurcations Del Negro lab (The College of William and Mary); Rubin, Hayes, Mendenhall, and Del Negro, PNAS, 2009

22 model schematic Rubin, Hayes, Mendenhall, and Del Negro, PNAS, 2009

23 model dynamics single self-coupled cell see also Dunmyre, Del Negro and R., JCNS, nd slow variable (Na + ) builds up

24 3) mixed-mode oscillations

25 Hodgkin-Huxley equations (1952)

26 MMOs in HH model change time scale parameter from 1 to 2 mechanism Rubin and Wechselberger, Biol. Cyb., 2008 (1) classical canards J. Moehlis Wechselberger, Scholarpedia

27 mechanism (2) generalize to R 3 jump up from fold as in bursting! (3) blow up the funnel Curtu and Rubin, in prep.

28 4) networks approaches: a) symmetry-based (Golubitsky, Buono, Leite, et al.) Theorem: the simplest CPG network for 2nlegged locomotion, satisfying reasonable assumptions, has 4nelements; moreover, for n=2, we have:

29 b) connectivity-based (Pecora, Belykh, Hasler, et al.) mostly for synchronization

30 c) brute force simulation of neuronal models (Prinz, Bucher, et al.) 2 very different parameter sets give ~same rhythm

31 average over 452,516 networks that give 3 classes of rhythms histograms of strengths of synapses

32 d) geometric dynamical systems approaches! Ex 1. synaptic excitation can powerfully boost bursting Butera et al., 1999 g tonic-e I tonic ~ f(v post ) I syn ~ g(v pre )f(v post )

33 burst promotion in more detail g tonic-e g syn-e Butera et al., 1999

34 Butera model but now we need 2 copies depend on v j

35 spike synchrony observed numerically to be unstable in all cases Best, Borisyuk, Rubin, Terman, and Wechselberger, SIADS, 2005

36 slow averaged dynamics example: g syn-e = 3 asymmetric bursting bursting solution (g ton-e = 0.83) symmetric bursting (g ton-e = 0.57) averaged nullclines oscillatory region jump-down curve asymmetric spiking (g ton-e = 0.87) symmetric spiking (g ton-e = 0.91)

37 g ton-e asymmetric bursting / enhanced duration g syn-e symmetric bursting asymmetric bursting asymmetric spiking symmetric spiking

38 Ex 2. transition mechanisms and feedback control half-center oscillator (Brown, 1911): components not intrinsically rhythmic; generates rhythmic activity, without rhythmic drive reciprocal inhibition active phase NOTE: spikes in active phase are omitted! silent phase

39 time courses for half-center oscillations from 3 mechanisms: persistent sodium, post-inhibitory rebound (T-current), adaptation (Ca/K-Ca)

40 simulation results: unequal constant drives fixed varied persistent sodium intermediate relative silent phase duration for cell with varied drive relative silent phase duration for cell with fixed drive post-inhibitory rebound adaptation Daun, Rubin, and Rybak, JCNS, 2009

41 Why? transition mechanisms: escape vs. release slow inhibition on inhibition off inhibition on inhibition off fast fast Wang & Rinzel, Neural Comp., 1992; Skinner et al., Biol. Cyb., 1994

42 Summary escape: independent phase modulation (e.g., persistent sodium current) release: poor phase modulation (e.g., postinhibitory rebound) adaptation = mix of release and escape: phase modulation NOT independent (e.g., Ca/K-Ca currents) Daun, Rubin, and Rybak, JCNS, 2009

43 Ex 3. what about larger respiratory networks (>2 slow variables)? (a) excitatory prebötc kernel is embedded within 3- component inhibitory ring network Rubin et al., J. Neurophysiol., 2009 Rubin et al., JCNS, 2010

44 (b) prebötc kernel itself is large and heterogeneous Gaiteri and Rubin, submitted

45 within network, determine when cells are bursting and then quantify network burstiness (NBI)

46 evaluate variation in NBI with network coupling architecture and cell type placement OP: analytical results on what ingredients determine burst synchronization in heterogeneous network? cf. Dunmyre & Rubin: tonic + quiescent; what about CAN bursters??

47 Ex 4. limbed (neuromechanical) locomotion model CPG (RGs, INs) motoneurons muscles + pendulum Markin et al., Ann. NY Acad. Sci., 2009 Spardy et al., SFN, 2010; J. Neural Eng., in prep

48 locomotion with feedback asymmetric phase modulation under variation of drive drive does this asymmetry imply asymmetry of CPG?

49 no! model has symmetric CPG yet still gives asymmetry if feedback is present locomotion with feedback asymmetric phase modulation under variation of drive drive locomotion without feedback loss of asymmetry drive Markin et al., Ann NY Acad Sci, 2009

50 rhythm with/without feedback: what is the difference? with feedback IN escape controls phase transitions Lucy Spardy

51 rhythm with/without feedback: what is the difference? without feedback RG escape controls phase transitions Lucy Spardy

52 idea: drive strength affects timing of INF escape (end of stance), RGE, RGF escape but not timing of INE escape drive research goal: show how drive differential equation model yields these results!

53 summary 1. some neurons/networks exhibit dynamically rich behavior called bursting 2. single cell bursting can be studied via fast/slow decomposition and bifurcation analysis 3. a classification of bursting in single cells exists but is arguably not complete (noise, multibif, MMOs) 4. bursting in networks can be studied in various ways 5. synaptic excitation promotes bursting by desynchronizing spikes: slow averaged dynamics 6. responses to feedback/drive depend on transition mechanisms within bursting rhythms 7. remains to determine key ingredients for synchronized bursting in heterogeneous networks

54 leech heart Cymbalyuk et al., J. Neurosci., 2002 respiratory CPG Rybak et al., Smith et al., Rubin et al.

55 Ex. 2 transition mechanisms determine responses to drive modulation step 1: eliminate spikes! Pace et al., Eur. J. Neurosci., 2007: prebötzinger Complex (mammalian respiratory brainstem)

56 asymmetric drive results (cont.) robust highly tunable high phase independence