Bursting Pacemaker Neurons

Based on: Models of Respiratory Rhythm Generation in the Pre-Botzinger Complex. I. Bursting Pacemaker Neurons Robert.J. Butera, John Rinzel, Jeffery C. Smith

Introduction

Pre-Bötzinger complex

 Pre-Bötzinger complex is the hypothesized site for respiratory rhythm generation  It is housed in the rostral ventrolateral medulla Part B, Photo labeling the ventrolateral medulla (pre-Bötzinger complex area approximated by dashed line).

Pacemaker Neurons

    Pre-Bötzinger complex houses the pacemaker neurons It is hypothesized that contribution of both a pacemaker-based kernel and a pattern-formatting network driven by the kernel is responsible for the respiratory rhythm generation (Hybrid model).

Some pacemaker neurons receive tonic excitatory inputs (from the mundane neurons) necessary to bring the membrane potential into the voltage window where bursting occurs.

These neurons are classified as conditional bursting pacemakers.

Background of the Research Paper

   In earlier models respiratory rhythm generation was postulated to arise from network interactions, specifically inhibitory connections. But in such models the rythmicity ceased when synaptic inhibition was blocked.

In the hybrid model, for which this paper is a segway, inhibitory interactions are not essential, mimics the actual in vitro and en bloc experiment results. The objective of this paper is modeling the rhythm and inspiratory burst generation in the kernel operating in vitro.

Model Development

Two models have been proposed for neurons responsible for rhythm and inspiratory burst generation in vitro.

Model 1

• Based on one-compartment Hodgkin-Huxley model. • Bursting occurs by virtue of fast activation and slow activation of a persistent Sodium current INa-P

Model 2

• Based on model 1. • Bursting occurs by virtue of fast-activating persistent Sodium current INa-P (inactivation term “h” removed) and slow activation of Potassium current IKs

Model 1

It is composed of five ionic currents across the plasma membrane: a fast sodium current, INa; a delayed rectifier potassium current, IK; a persistent sodium current, INaP; a passive leakage current, IL;and a tonic current, Itonic_e (although this last current is considered to be inactive in these models)

Model 1 - Formulation

C dV dt

 

I NaP



I Na



I K



I L



I tonic



e



I app

Where, x ∞ Є {mP,m,h,n} and x Є {h,n}

dx dt



x



(

V



x

(

V

) )



x



x



(

V

)

x

 

x



1 /{ 1



exp[(

V

/ cosh[(

V

 

x

) /



x

)]}

 

x

) /( 2



x

)]

Model 2

The second model is identical to the model 1 except with the addition of a slow K+ current, IKS. (The removal of the inactivation term "h" from INaP is not visible in the model diagram.)

Model 2 - Formulation

C dV dt

 

I NaP



I Na



I K



I KS



I L



I tonic



e



I app

Where, x ∞ Є {mP,m,k,n} and x Є {k,n}

dx dt



x



(

V



x

(

V

) )



x



x



(

V

)

x

 

x



1 /{ 1



exp[(

V

/ cosh[(

V

 

x

) /



x

)]}

 

x

) /( 2



x

)]

How does model 1 work?

Dynamic response of model 1 as a function of EL: membrane potential

EL=-60mV 20 0 -20 EL=-57.5 A Closer Look 20 -40 -60 0 1000 2000 3000 4000 5000 6000 7000

EL=-60 mV

8000 9000 10000 0 0.75

0.7

-20 -40 0.65

-60 0 0.6

100 200 300 400 500 600 700 800 900 1000 0.55

0 1 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0.5

0 0 1 1000 2000 3000 4000 5000 t (s) 6000 Phase plae plot n vs V 7000 8000 9000 10000 0.54

0.52

0.5

0.48

0.46

0.44

0 100 200 800 900 1000 300 400 500 t (s) 600 700

A closer look…

0.5

0 -60 -50 -40 -30 v (V) -20 -10 0 10

Nullclines - m

∞ 3 , n ∞ 4 , h ∞

Nullclines minf, ninf, hinf 1 0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 -100 -80 -60 -40 -20

V (mV)

0 minf 3 ninf 4 hinf 20 40

EL=-57.5 mV

EL=57.5mV

20 0 -20 -40 -60 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0.54

0.52

0.5

0.48

0.46

0.44

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 function n vs t 1 0.5

0 0 1000 2000 3000 4000 5000 6000 t (s) Phase plane plot n vs V 7000 8000 9000 10000 1 0.8

0.6

0.4

0.2

0 -60 -50 -40 -30 v (V) -20 -10 0 10 0.5

0.7

0.6

0.5

0.4

EL=-54 mV

EL=-54mV 20 0 -20 -40 -60 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 1 0 1000 2000 3000 4000 5000 t (s) EL=-54mV 6000 7000 8000 9000 10000 0 0 1 0.8

0.6

0.4

0.2

0 -60 1000 2000 3000 4000 5000 6000 t (s) Phase plane plot 7000 8000 9000 10000 -50 -40 -30 v (V) -20 -10 0 10

Results from research article.

Bursting to tonic spiking

EL=-60 20 0 -20 -40 -60 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 -20 -40 -60 80 60 40 20 0 0.75

0.7

0.65

0.6

0.55

0 1000 2000 3000 4000 5000 t (s) ISub vs h 6000 7000 8000 9000 10000 120 100 0.58

0.59

0.6

0.61

0.62

0.63

0.64

0.65

0.66

0.67

EL=-57.5

20 0 -20 -40 -60 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 EL=-54 20 0 -20 -40 -60 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 40 20 0 -20 -40 -60 0.45

140 120 100 80 60 0.7

0.65

0.6

0.55

0.5

0.45

0 1000 2000 3000 4000 5000 t (s) 6000 7000 8000 9000 10000 ISub vs h 0.5

0.55

0.6

0.65

0.7

0.7

0.6

0.5

0.4

140 0 1000 2000 3000 4000 5000 t (s) 6000 7000 8000 9000 10000 ISub vs h 120 100 80 60 40 20 0 -20 -40 -60 0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

EL=-60 mV EL=-57.5 mV EL=-54 mV

Model 1- Animation

The two kinds of currents in this model are: 1) Spike generating currents - INa & IK 2) Sub threshold currents (INaP+ IL called Isub).

120 100 80 • • • •

The bursting cycle can be understood like this: When gNaP increases beyond a critical value, Isub is large enough to initiate a burst. The firing of action potentials gradually inactivates h (slow variable) The bursting terminates when INaP is inactivated sufficiently and the cell hyperpolarizes.

Now h gradually de-inactivates increasing Isub, to trigger another burst and so on…

60 40 20 0 -20 -40 -60 0 Isub vs Time burst-iv.qt.mov

200 400 600 800 1000 Time (s) 1200 1400 1600 1800 2000

Isub vs Time

Bifurcation Mechanism

EL=-65 mV Silent EL=-58 mV Bursting

SN bif. - saddle-node bifurcation HC bif - saddle-homoclinic bifurcation subH - subcritical Andronov-Hopf bifurcation.

EL=-55 mV Beating

0.5

Model 2 -Results

EL=-59.5 mV 20 0 -20 -40 -60 -80 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 EL=-50 mV 20 0 -20 -40 -60 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0.14

0.12

0.1

0.08

0.06

0.04

0 1000 2000 3000 4000 5000 t (s) 6000 7000 8000 9000 10000 nvs t and phase plane plot 1 0.3

0.25

0.2

0.15

0.1

0.05

0 1000 2000 3000 4000 5000 t (s) 6000 7000 8000 9000 10000 n vs t and phase plot EL=-50mV 1 0.5

0 0 1000 2000 3000 4000 5000 t (s) 6000 7000 8000 9000 10000 1 0.8

0.6

0.4

0.2

0 -70 -60 -50 -40 -30 v (V) -20 -10 0 10

EL=-59.5 mV

0 0 1000 2000 3000 4000 5000 t (s) 6000 7000 8000 9000 10000 1 0.8

0.6

0.4

0.2

0 -60 -50 -40 -30 v (V) -20 -10 0 10

EL=-50 mV

EL=-40mV 20 0 -20 -40 -60 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0.4

0.35

0.3

0.25

0.2

1 0 1000 2000 3000 4000 5000 t (s) 6000 7000 8000 9000 10000 n vs t and phase plot EL=-40mV 0.5

0 0 1000 2000 3000 4000 5000 t (s) 6000 7000 8000 9000 10000 1 0.8

0.6

0.4

0.2

0 -60 -50 -40 -30 v (V) -20 -10 0 10

EL=-40 mV

Results from research article

Model 2 -working



Model 2 operates in a very similar fashion as Model 1, the difference being the slow activation persistent sodium current INaP is replaces by a slow activation of potassium current IKS

Difference between Models 1 & 2

1 2

     Burst initiated by INaP, terminated by inactivating INaP Membrane conductance gm increases through the silent phase The membrane potential remains flat during the inter burst interval Burst duration decreases with depolarization Supports bursting over a small range of EL (-60 to -54 mV)      Burst initialed by INaP, terminated by activating IKS sufficiently Membrane conductance gm decreases through the silent phase The membrane potential interval not as flat during the inter burst interval Burst duration does not decrease with depolarization Supports bursting over double the range of EL as model 1 (-59.5 to -40 mV)

Both models go through a regime of silence bursting and beating

In both models a minimum value of gNaP is required to support bursting. If gNaP is too low, only quiescence , or for higher values beating are supported.

Miscellaneous Comments

 The same effects of chaging EL can be obtained by fixing EL and varying the parameter gtonic (Itonic) or Iapp ( gL (v-EL) in model 2). Some of these results shown below.

Summary

    Model 1 is found to be more consistent with experimental data. The relative flat interburst interval is due to the fact that the subthreshold currents are all balanced and add up to zero. These are minimal models that provide a believable explanation for generating multistate, voltage-dependent behavior observed in the Pre-Botzinger pacemaker neurons.

Although the actual burst generating currents still need to be unidentified in the Pre-Botzinger neurons

Questions?

Happy Holidays

Neuron Xmas tree!!

References

  RT-PCR reveals muscarinic acetylcholine receptor mRNA in the pre-Bötzinger complex, Jiunu Lai * , Xuesi M. Shao * , Richard W. Pan, Edward Dy, Cindy H. Huang, and Jack L. Feldman Models of Respiratory Rhythm Generation in the Pre-Bo¨tzinger Complex. I. Bursting Pacemaker   Neurons, ROBERT J. BUTERA, JR.,1,2 JOHN RINZEL,1–3 AND JEFFREY C. SMITH1 The Dynamic Range of Bursting in a Model Respiratory Pacemaker Network , Janet Best, Alla Borisyuk, Jonathan Rubin, David Terman, Martin Wechselberger All simulations performed using Matlab 7.0 , with a ode15s solver and absolute and relative tolerance of 10-6.