department of economics / computational neuroeconomics group neural adaptation and bursting or: a...

Department of Economics / Computational Neuroeconomics Group

Neural Adaptation and Burstingor: A dynamical taxonomy of neurons

April 27th, 2011

Lars Kasper


Introduction and Link to last sessions


Chapter 10 – Neural Adaptation and Bursting

Symbols & Numbers

V membrane potential

R recovery variable (related to K+)

H conductance variable (related to slow K+ current, IAHP)

C very slow K+ (IAHP) conductance mediated by intracellular Ca2+ concentration

X Ca2+ conductance, rapid depolarizing current

IA rapid transient K+ current

IAHP slow afterhyperpolarizing K+ current

IADP slow afterdepolarizing current (fast R and slow X comb.)

+55, +48 mV Na+ equilibrium potential

+140 mV Ca2+ equilibrium potential

-95, -92 mV K+ equilibrium potential

-70, -75.4 mV Resting membrane potential

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Overview of Introduced Neuron ModelsModel Hodgin-Huxley/Rinzel Connor et al.

Rose&Hindmarsh

Neuron type Class II (squid axon) Class I (fast-spiking, inhib. cortical neuron)

Experimental phenomena explained

• High frequency firing (175-400 Hz)

• High and low frequency firing (1-400 Hz)

Included Ion Currents • Depolarizing Na+

(fast)• Hyperpolarizing K+

(slow)

• Depolarizing Na+• Hyperpolarizing K+

• Transient Hyper-polarizing K+ (fast)

Dynamical system characteristics

• Hard Hopf bifurcation• => hysteresis of

cease-fire current

• Saddle-node bifurcation

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Take home message: More fun with currents

• Essentially deepest insight of today’s session: Spike frequency and AP creation are dependent on external, stimulating current.

• Today some intrinsic currents will partially counteract the effect of the external driving current.

• This will be done in a dynamic manner via the introduction of 1 or 2 additional currents modelling

• Afterhyperpolarizing effects (very slow K+)

• Additional depolarizing effects (fast Ca2+)

• This dynamic net current fluctuation will lead to complex behavior due to recurring back- and forth-crossings of bifurcation boundaries

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Today: Completing the single neuron taxonomy

• Fast-spiking inhibitory neurons• Regular-spiking excitatory neurons

• with spike rate adaptation• Current-driven bursting neurons

• Chattering neurons

Class I (mammalian)

• Fast-spiking neurons• Endogenous bursting neurons

Class II (squid/invertebrate)

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Topics

• Introduction and scope• There’s much more to neurons than spiking

• Spike frequency adaptation• Neural bursting and hysteresis

• Class II Neurons• Endogenous bursting

• Class I neurons• Separating limit cycles using a neurotoxin

• Constant current-driven bursting • Neocortical neurons

• Summary: The neuron model zoo


Spike Frequency Adaptation

• What is spike rate adaptation?• Threefold reduction of spike rates within

100 ms of constant stimulation typical for cortical neurons

• Which current is introduced?• Very slow hyperpolarizing K+ current• Mediated by Ca2+ influx

• What function does it enable?• Short-term memory• Neural competition


Spike Rate Adaptation70 Hz 25 Hz



Recap: Rinzel-model with transient K+ current

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Transient K+ via quadratic voltage-dependence of recovery

Voltage V

Recovery variable R



After-hyperpolarization via slow K+ current

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H: Conductance of slow K+ (after-)hyperpolarizing current IAHP

No resting state effect

K+



Explanation via reduction of effective driving current

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Simulation: RegularSpiking.m with I=0.85, 1.8

• H has no effect on action potential (slow time constant)• H is driven by supra-threshold voltages

• Then counteracts driving current in dV/dt



Capability of the model

4/27/2011

• Predicts current-independent threefold reduction in spike rate from transient to steady state

• Predicts linear dependence of spike rates on input current

• But: fails to explain high-current saturation effects

• Voltage dependent recovery time constant of R needed

• Pharmacological intervention model: IAHP can be blocked or reduced by neuromodulators (ACh, histamine, norepinephrine, serotonin)



Wrap-up: Completing the single neuron taxonomy




Class I (mammalian)



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Neural Bursting and Hysteresis – Class II neurons

• What is Bursting?• Short train of several spikes interleaved with

phases of silence• Which current is introduced?

• Might be the same as for spike rate adaptation• Very slow hyperpolarizing K+ current

• What function does it enable?• Complex behavioral change of network• Synchronization• “Multiplexing”: driving freq-specific neurons


Slow hyperpolarization in a squid axon

Standard Class II neuron:

Class II neuron with slow hyperpolarization IAHP due to K+ current:



Bursting Neurons

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Simulation: HHburster.m with I=0.14, 0.18



Bursting Neurons

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Simulation: HHburster.m with I=0.14, 0.18

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

V-R projection of phase space trajectories (red)



Bursting analysis of bifurcation diagram

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Bursting analysis of bifurcation diagram

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Inet ↑

H ↑

V ↑

V ↓

Inet ↓

H ↓

𝑑𝑉𝑑 𝑡

∝ 𝐼𝑛𝑒𝑡

𝑑𝐻𝑑𝑡

∝9.3 (𝑉 −𝑉 𝑟𝑒𝑠𝑡 )

𝐼𝑛𝑒𝑡=𝐼−0.54𝐻 ¿

Actionpotential

(b)

(a)

(c)

APvanishes

(d)



Bursting Analysis of Bifurcation diagram

4/27/2011

(a)

(b)

(c)

(d)


Endogenous Bursting

Californian Aplysia (Seehase)

• Rinzel model for Class I – neurons• More realistic 4-current model


Endogenous Bursting

• What is endogenous bursting?• Occurrence of bursting neuronal activity in the

absence of external stimulation (via a current I)• Which currents are introduced?

• Fast depolarizing Ca2+-influx conductance X• Slow hyperpolarizing K+ conductance C

• What function does it enable?• Pacemaker neurons (heartbeat, breathing)• synchronization


A more complex model of 4 intrinsic currents“Plant-model”

• X is voltage-dependent (voltage-gated Ca2+ channels)• C is Ca2+-concentration dependent (Ca2+-activated K+ channels)• No external currents occur



Comparison to 3-current model of spike rate adaptation

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Now termed C, IAHP



Endogenous Bursting Neuron: in-vivo

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Difference to former model:• No stimulating current• Modulation back- and forth a saddle-node bifurcation



Endogenous Bursting Neuron: in silico

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0 20 40 60 80 100 120 140 160 180 200-80

-60

-40

-20

0

20

40

Time (ms)

Enlargement of burst between 400-700 ms

Simulation: PlantBurster.m

0 200 400 600 800 1000 1200 1400 1600-80

-70

-60

-50

-40

-30

-20

-10

0

10

20

Time (ms)

Pot

entia

l (m

V)

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

1.2

1.4

C

X

X-C Projection of Phase Space

X-C-projection ofPhase space

• Burst phases again occur due to a crossing of a bifurcation point enabling a limit cycle

• Due to Rinzel model: saddle node bifurcation

• Additional currents X&C follow a limit cycle themselves with slower time scale than V-R (visible as ripples in projection)

Time course of voltage V







Class I (mammalian)



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Separating limit cycles via intoxication

Californian Aplysia (Seehase) Puffer Fish (Kugelfisch)

VS



Tetrodotoxin and Sushi

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• Tetrodotoxin (TTX) acts as nerve poison via blocking of the depolarizing Na+ channels

• Neurons cannot create action potentials any longer

Removed voltage dependency of Na+ conductance



Silencing all Na+-channels – in vivo

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Silencing all Na+-channels: in silico

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Without TTX

• Still fluctuation due to X-C dynamics

• No action potentials created

With TTX



Remaining limit cycle without Na+ current

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Simulation: PlantBursterTTX.m

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

C

X


0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

1.2

1.4

C

X


Without TTX With TTX

• X-C-projection of Phase space exhibits same limit cycle behavior• Modulation of X due to voltage changes vanish


Current-driven Bursting in Neocortical Neurons

• What is endogenous bursting?• Occurrence of bursting neuronal activity in

response to a constant external stimulation (via a current I)

• Which currents are introduced?• External, stimulating current I• Fast depolarizing Ca2+-influx conductance X• Slow hyperpolarizing K+ conductance C

• What function does it enable?• Chattering sensory neurons



Sensory cell bursting

4/27/2011

Mouse somatosensory cortex neuron Cat visual cortex neuron



Driving Current

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Driving Current: differences to endogenous bursting model

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Driven bursting in a neocortical neuron

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• Hopf bifurcation of X-C at I=0.197• Qualitatively similar behavior of X-C limit cycle above

this threshold to endogenous spiking• X-C limit-cycle drives V-R subspace through saddle-

node bifurcation• One limit cycle driving the other to create bursts• But not autonomous due to V-dependence of X

Simulation: Chattering.m

0 50 100 150 200 250 300 350 400 450-76

-74

-72

-70

-68

-66

Time (ms)

Pote

ntial (m

V) I=0.19

0 50 100 150 200 250 300 350 400 450 500-80

-70

-60

-50

-40

-30

-20

-10

0

10

20

Time (ms)

Pote

ntial (m

V) I=0.2

-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

C

X


I=0.2

X-C-projection of phase spaceTime course of voltage V







Class I (mammalian)



4/27/2011



Dynamical Taxonomy of Class I neurons

4/27/2011

Fast-SpikingInhibitory interneurons

Regular SpikingExcitatory Neurons

• Only 2 ion channel currents (Rinzel-model)• fast Na+ depolarization• slow K+ recovery

• Constant spike rate: 1-400 Hz

• Additional 3rd current• very slow after-hyperpolarizing K+ current

• Enables spike rate adaptation

Neocortical Bursting Cells

• Additional 3rd & 4th current• very slow after-hyperpolarizing K+ current,

mediated by Ca2+ concentration• fast depolarizing Ca2+ current

• Enables bursting, either intrinsic () as pacemaker or driven by an external current



Dynamical Taxonomy of Class I neurons

4/27/2011

Fast-SpikingInhibitory interneurons

Regular SpikingExcitatory Neurons

Neocortical Bursting Cells

𝑑𝑉𝑑 𝑡

=− 𝑓 11 (𝑉 2 ) ⋅¿ 𝑑𝑉𝑑 𝑡

=…− 𝑓 13 (1 ) 𝐻 ¿ 𝑑𝑅𝑑𝑡

=…

𝑑𝐻𝑑𝑡

=1𝜏𝐻

(−𝐻+ 𝑓 31 (𝑉 ) (𝑉 −𝑉 𝑟𝑒𝑠𝑡 ))with𝜏𝐻≫𝜏 𝑅

𝑑𝑉𝑑 𝑡

=…− 𝑓 14 (1 ) 𝑋 ¿ 𝑑𝑅𝑑𝑡

=…

𝑑𝐻𝑑𝑡

=1𝜏𝐻

(−𝐻+ 𝑓 31 (1 ) 𝑋 )𝑑 𝑋𝑑𝑡

=1𝜏 𝑋

(− 𝑋+ 𝑓 31 (𝑉 ) (𝑉 −𝑉 𝑟𝑒𝑠𝑡 ))with𝜏𝐻≫𝜏 𝑋>𝜏𝑅

𝐼 𝐴𝐻𝑃

𝐼𝑇



Take home message: More fun with currents

• Spike frequency and AP creation are dependent on external, stimulating current.

• Intrinsic currents partially counteract the effect of the external driving current.

• This happens in a dynamic manner via the introduction of 1 or 2 additional currents modelling

• Afterhyperpolarizing effects (very slow K+)

• Additional depolarizing effects (fast Ca2+)

• This dynamic net current fluctuation leads to complex behavior due to recurring back- and forth-crossings of bifurcation boundaries

4/27/2011



Picture Sources

http://upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Tetrodotoxin.svg/1000px-Tetrodotoxin.svg.png

http://upload.wikimedia.org/wikipedia/commons/7/77/Puffer_Fish_DSC01257.JPG

http://upload.wikimedia.org/wikipedia/commons/e/ef/Aplysia_californica.jpg

http://www.cvr.yorku.ca/webpages/spikes.pdf => Chapter 9 and 10

4/27/2011








http://www.cvr.yorku.ca/webpages/spikes.pdf

http://www.cvr.yorku.ca/webpages/spikes.pdf

department of economics / computational neuroeconomics group neural adaptation and bursting or: a...

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