The dynamics of bursting in neurons and networks

Professor Jonathan Rubin January 20, 2011

Outline:

• Introduction to neurons

• Introduction to bursting

• Mathematical analysis of bursting in single cells

• Frontiers in research on bursting dynamics (21-53)

a cartoon neuron

4-100 µm in diameter, ~10-6 grams each; ~25,000 m2 surface area over whole human brain (like 4 soccer fields)

patch-clamp recordings (CNRS, France)

action potential generation

cell membrane

OUTSIDE

INSIDE

examples of bursting in neurons (Scholarpedia)

•  multiple action potentials at relatively high frequency (100 Hz)

•  long interburst intervals

why care about bursting?

1.  mathematically interesting

2.  plays a role in the brain

(a) sleep

(b) novelty detection

(c) central pattern generators (CPGs)

crustacean STG – Rabbeh and Nadim, J. Neurophysiol., 2007

Nat. Rev. Neurosci., 2005

(d) and pathology Raz et al., J. Neurosci., 2000

example: the respiratory pre-Bötzinger complex (preBötC)

embedded in network

isolated

Butera, Rinzel, Smith, 1999

mathematical modeling/analysis: single-neuron bursting

time

voltage

bursting in differential equation models

periodic

chaotic

Butera et al. model

Butera, Rinzel, Smith, 1999

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

slow inactivation of persistent sodium (h)

fast/slow decomposition for the Butera et al. model

deinactivation of persistent sodium (h)

in this example, there is a stable fixed point corresponding to quiescence slow

fast

gtonic-e A: quiescent B,C: bursting

D: tonic spiking

conditional square-wave bursting in the Butera et al. model

rigorous framework – Terman, 1991-2

full classification? Izhikevich, 2000 (pg. 1171-1266) sl

ow

fast

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

1) noise

Kuske and Baer, Bull. Math. Biol., 2002

Su, R. and Terman, Nonlinearity, 2004

Pedersen and Sorensen, SIAP, 2007

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

2) multiple bifurcations

model schematic

Rubin, Hayes, Mendenhall, and Del Negro, PNAS, 2009

model dynamics – single self-coupled cell

2nd slow variable (Na+) builds up see also

Dunmyre, Del Negro and R., JCNS, 2011

3) mixed-mode oscillations

Hodgkin-Huxley equations (1952)

MMOs in HH model – change time scale parameter from 1 to 2

(1) classical canards

mechanism

Rubin and Wechselberger, Biol. Cyb., 2008

J. Moehlis Wechselberger, Scholarpedia

(2) generalize to R3

mechanism

jump up from fold – as in bursting!

(3) blow up the funnel

Curtu and Rubin, in prep.

4) networks approaches:

a) symmetry-based (Golubitsky, Buono, Leite, et al.)

Theorem: the simplest CPG network for 2n-legged locomotion, satisfying reasonable assumptions, has 4n-elements; moreover, for n=2, we have:

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

c) brute force simulation of neuronal models (Prinz, Bucher, et al.)

2 very different parameter sets give ~same rhythm

average over 452,516 networks that give 3 classes of rhythms

histograms of strengths of synapses

Ex 1. synaptic excitation can powerfully boost bursting

gtonic-e

Butera et al., 1999

Isyn ~ g(vpre)f(vpost)

Itonic ~ f(vpost)

d) geometric dynamical systems approaches!

burst promotion in more detail

Butera et al., 1999

gtonic-e

gsyn-e

depend on vj

Butera model – but now we need 2 copies

•  spike synchrony observed numerically to be unstable in all cases

Best, Borisyuk, Rubin, Terman, and Wechselberger, SIADS, 2005

slow averaged dynamics example: gsyn-e = 3

(gton-e = 0.57)

(gton-e = 0.91) (gton-e = 0.87)

asymmetric bursting (gton-e = 0.83) bursting solution

averaged nullclines

jump-down curve

oscillatory region

symmetric bursting

asymmetric spiking

symmetric spiking

symmetric bursting asymmetric bursting asymmetric spiking symmetric spiking

gton-e

gsyn-e

asymmetric bursting / enhanced duration

half-center oscillator (Brown, 1911): components not intrinsically rhythmic; generates rhythmic activity, without rhythmic drive

reciprocal inhibition

−

−

silent phase

active phase

Ex 2. transition mechanisms and feedback control

NOTE: spikes in active phase are omitted!

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

simulation results: unequal constant drives

intermediate

adaptation

persistent sodium

post-inhibitory rebound

relative silent phase duration for cell with varied drive

relative silent phase duration for cell with fixed drive

Daun, Rubin, and Rybak, JCNS, 2009

fixed varied −

−

Why? transition mechanisms: escape vs. release

Wang & Rinzel, Neural Comp., 1992; Skinner et al., Biol. Cyb., 1994

inhibition on

inhibition off

inhibition on inhibition off

fast fast

slow

Summary

• escape: independent phase modulation (e.g., persistent sodium current)

•  release: poor phase modulation (e.g., post-inhibitory rebound)

•  adaptation = mix of release and escape: phase modulation NOT independent (e.g., Ca/K-Ca currents)

Daun, Rubin, and Rybak, JCNS, 2009

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

(b) preBötC kernel itself is large and heterogeneous

Gaiteri and Rubin, submitted

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

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??

Ex 4. limbed (neuromechanical) locomotion model

Markin et al., Ann. NY Acad. Sci., 2009

Spardy et al., SFN, 2010; J. Neural Eng., in prep

CPG (RGs, INs)

motoneurons

muscles + pendulum

drive

locomotion with feedback – asymmetric phase modulation under variation of drive

does this asymmetry imply asymmetry of CPG?

locomotion with feedback – asymmetric phase modulation under variation of drive

locomotion without feedback – loss of asymmetry

drive

drive

no! – model has symmetric CPG yet still gives asymmetry if feedback is present

Markin et al., Ann NY Acad Sci, 2009

rhythm with/without feedback: what is the difference?

with feedback

IN escape controls phase transitions

Lucy Spardy

without feedback

RG escape controls phase transitions

Lucy Spardy

rhythm with/without feedback: what is the difference?

drive

drive

idea: drive strength affects timing of INF escape (end of stance), RGE, RGF escape but not timing of INE escape

research goal: show how differential equation model yields these results!

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

summary

leech heart – Cymbalyuk et al., J. Neurosci., 2002

respiratory CPG – Rybak et al., Smith et al., Rubin et al.

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)

asymmetric drive results (cont.)

highly tunable

high phase independence

robust