Department of Economics / Computational Neuroeconomics Group
Neural Adaptation and Burstingor: A dynamical taxonomy of neurons
April 27th, 2011
Lars Kasper
Department of Economics / Computational Neuroeconomics Group
Introduction and Link to last sessions
Department of Economics / Computational Neuroeconomics Group
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
4/27/2011 Page 3
Department of Economics / Computational Neuroeconomics Group
Chapter 10 – Neural Adaptation and Bursting
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
4/27/2011 Page 4
Department of Economics / Computational Neuroeconomics Group
Chapter 10 – Neural Adaptation and Bursting
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
4/27/2011 Page 5
Department of Economics / Computational Neuroeconomics Group
Chapter 10 – Neural Adaptation and Bursting
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)
4/27/2011 Page 6
Department of Economics / Computational Neuroeconomics Group
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
Department of Economics / Computational Neuroeconomics Group
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
Department of Economics / Computational Neuroeconomics Group
Spike Rate Adaptation70 Hz 25 Hz
Department of Economics / Computational Neuroeconomics Group
Chapter 10 – Neural Adaptation and Bursting
Recap: Rinzel-model with transient K+ current
4/27/2011 Page 10
Transient K+ via quadratic voltage-dependence of recovery
Voltage V
Recovery variable R
Department of Economics / Computational Neuroeconomics Group
Chapter 10 – Neural Adaptation and Bursting
After-hyperpolarization via slow K+ current
4/27/2011 Page 11
H: Conductance of slow K+ (after-)hyperpolarizing current IAHP
No resting state effect
K+
Department of Economics / Computational Neuroeconomics Group
Chapter 10 – Neural Adaptation and Bursting
Explanation via reduction of effective driving current
4/27/2011 Page 12
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
Department of Economics / Computational Neuroeconomics Group
Chapter 10 – Neural Adaptation and Bursting
Capability of the model
4/27/2011 Page 13
• 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)
Department of Economics / Computational Neuroeconomics Group
Chapter 10 – Neural Adaptation and Bursting
Wrap-up: 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)
4/27/2011 Page 14
Department of Economics / Computational Neuroeconomics Group
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
Department of Economics / Computational Neuroeconomics Group
Slow hyperpolarization in a squid axon
Standard Class II neuron:
Class II neuron with slow hyperpolarization IAHP due to K+ current:
Department of Economics / Computational Neuroeconomics Group
Chapter 10 – Neural Adaptation and Bursting
Bursting Neurons
4/27/2011 Page 17
Simulation: HHburster.m with I=0.14, 0.18
Department of Economics / Computational Neuroeconomics Group
Chapter 10 – Neural Adaptation and Bursting
Bursting Neurons
4/27/2011 Page 18
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)
Department of Economics / Computational Neuroeconomics Group
Chapter 10 – Neural Adaptation and Bursting
Bursting analysis of bifurcation diagram
4/27/2011 Page 19
Department of Economics / Computational Neuroeconomics Group
Chapter 10 – Neural Adaptation and Bursting
Bursting analysis of bifurcation diagram
4/27/2011 Page 20
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)
Department of Economics / Computational Neuroeconomics Group
Chapter 10 – Neural Adaptation and Bursting
Bursting Analysis of Bifurcation diagram
4/27/2011 Page 21
(a)
(b)
(c)
(d)
Department of Economics / Computational Neuroeconomics Group
Endogenous Bursting
Californian Aplysia (Seehase)
• Rinzel model for Class I – neurons• More realistic 4-current model
Department of Economics / Computational Neuroeconomics Group
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
Department of Economics / Computational Neuroeconomics Group
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
Department of Economics / Computational Neuroeconomics Group
Chapter 10 – Neural Adaptation and Bursting
Comparison to 3-current model of spike rate adaptation
4/27/2011 Page 25
Now termed C, IAHP
Department of Economics / Computational Neuroeconomics Group
Chapter 10 – Neural Adaptation and Bursting
Endogenous Bursting Neuron: in-vivo
4/27/2011 Page 26
Difference to former model:• No stimulating current• Modulation back- and forth a saddle-node bifurcation
Department of Economics / Computational Neuroeconomics Group
Chapter 10 – Neural Adaptation and Bursting
Endogenous Bursting Neuron: in silico
4/27/2011 Page 27
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
Department of Economics / Computational Neuroeconomics Group
Chapter 10 – Neural Adaptation and Bursting
Wrap-up: 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)
4/27/2011 Page 28
Department of Economics / Computational Neuroeconomics Group
Separating limit cycles via intoxication
Californian Aplysia (Seehase) Puffer Fish (Kugelfisch)
VS
Department of Economics / Computational Neuroeconomics Group
Chapter 10 – Neural Adaptation and Bursting
Tetrodotoxin and Sushi
4/27/2011 Page 30
• 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
Department of Economics / Computational Neuroeconomics Group
Chapter 10 – Neural Adaptation and Bursting
Silencing all Na+-channels – in vivo
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Department of Economics / Computational Neuroeconomics Group
Chapter 10 – Neural Adaptation and Bursting
Silencing all Na+-channels: in silico
4/27/2011 Page 32
Without TTX
• Still fluctuation due to X-C dynamics
• No action potentials created
With TTX
Department of Economics / Computational Neuroeconomics Group
Chapter 10 – Neural Adaptation and Bursting
Remaining limit cycle without Na+ current
4/27/2011 Page 33
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
X-C Projection of Phase Space
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
Without TTX With TTX
• X-C-projection of Phase space exhibits same limit cycle behavior• Modulation of X due to voltage changes vanish
Department of Economics / Computational Neuroeconomics Group
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
Department of Economics / Computational Neuroeconomics Group
Chapter 10 – Neural Adaptation and Bursting
Sensory cell bursting
4/27/2011 Page 35
Mouse somatosensory cortex neuron Cat visual cortex neuron
Department of Economics / Computational Neuroeconomics Group
Chapter 10 – Neural Adaptation and Bursting
Driving Current
4/27/2011 Page 36
Department of Economics / Computational Neuroeconomics Group
Chapter 10 – Neural Adaptation and Bursting
Driving Current: differences to endogenous bursting model
4/27/2011 Page 37
Department of Economics / Computational Neuroeconomics Group
Chapter 10 – Neural Adaptation and Bursting
Driven bursting in a neocortical neuron
4/27/2011 Page 38
• 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
X-C Projection of Phase Space
I=0.2
X-C-projection of phase spaceTime course of voltage V
Department of Economics / Computational Neuroeconomics Group
Chapter 10 – Neural Adaptation and Bursting
Wrap-up: 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)
4/27/2011 Page 39
Department of Economics / Computational Neuroeconomics Group
Chapter 10 – Neural Adaptation and Bursting
Dynamical Taxonomy of Class I neurons
4/27/2011 Page 40
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
Department of Economics / Computational Neuroeconomics Group
Chapter 10 – Neural Adaptation and Bursting
Dynamical Taxonomy of Class I neurons
4/27/2011 Page 41
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
Department of Economics / Computational Neuroeconomics Group
Chapter 10 – Neural Adaptation and Bursting
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 Page 42
Department of Economics / Computational Neuroeconomics Group
Chapter 10 – Neural Adaptation and Bursting
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 Page 43
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