Chapter 10
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Department of Economics / Computational Neuroeconomics Group Neural Adaptation and Bursting or: 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 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 Chapter 10 – Neural Adaptation and Bursting Page 3 Department of Economics / Computational Neuroeconomics Group Overview of Introduced Neuron Models Model 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 (1400 Hz) Included Ion Currents • Depolarizing Na+ (fast) • Hyperpolarizing K+ (slow) • Depolarizing Na+ • Hyperpolarizing K+ • Transient Hyperpolarizing K+ (fast) Dynamical system characteristics • Hard Hopf bifurcation • Saddle-node • => hysteresis of bifurcation cease-fire current 4/27/2011 Chapter 10 – Neural Adaptation and Bursting Page 4 Department of Economics / Computational Neuroeconomics Group 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 Chapter 10 – Neural Adaptation and Bursting Page 5 Department of Economics / Computational Neuroeconomics Group Today: Completing the single neuron taxonomy Class I (mammalian) Class II (squid/invertebrate) 4/27/2011 • Fast-spiking inhibitory neurons • Regular-spiking excitatory neurons • with spike rate adaptation • Current-driven bursting neurons • Chattering neurons • Fast-spiking neurons • Endogenous bursting neurons Chapter 10 – Neural Adaptation and Bursting 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 Adaptation Department of Economics / Computational Neuroeconomics Group Recap: Rinzel-model with transient K+ current 4/27/2011 Chapter 10 – Neural Adaptation and Bursting Page 10 Department of Economics / Computational Neuroeconomics Group After-hyperpolarization via slow K+ current 4/27/2011 Chapter 10 – Neural Adaptation and Bursting Page 11 Department of Economics / Computational Neuroeconomics Group Explanation via reduction of effective driving current 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 4/27/2011 Chapter 10 – Neural Adaptation and Bursting Page 12 Department of Economics / Computational Neuroeconomics Group Capability of the model • 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) 4/27/2011 Chapter 10 – Neural Adaptation and Bursting Page 13 Department of Economics / Computational Neuroeconomics Group Wrap-up: Completing the single neuron taxonomy Class I (mammalian) Class II (squid/invertebrate) 4/27/2011 • Fast-spiking inhibitory neurons • Regular-spiking excitatory neurons • with spike rate adaptation • Current-driven bursting neurons • Chattering neurons • Fast-spiking neurons • Endogenous bursting neurons Chapter 10 – Neural Adaptation and Bursting 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 Bursting Neurons Simulation: HHburster.m with I=0.14, 0.18 4/27/2011 Chapter 10 – Neural Adaptation and Bursting Page 17 Department of Economics / Computational Neuroeconomics Group Bursting Neurons Simulation: HHburster.m with I=0.14, 0.18 V-R projection of phase space trajectories (red) 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 -1 4/27/2011 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Chapter 10 – Neural Adaptation and Bursting 0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Page 18 Department of Economics / Computational Neuroeconomics Group Bursting analysis of bifurcation diagram 4/27/2011 Chapter 10 – Neural Adaptation and Bursting Page 19 Department of Economics / Computational Neuroeconomics Group Bursting analysis of bifurcation diagram + πΎ ) πΌπππ‘ = πΌ − 0.54π»(π − πππ ππ ∝ πΌπππ‘ ππ‘ Inet ↑ V↑ Action potential H↓ ππ» ∝ 9.3 π − ππππ π‘ ππ‘ H↑ V↓ AP vanishes Inet ↓ 4/27/2011 Chapter 10 – Neural Adaptation and Bursting Page 20 Department of Economics / Computational Neuroeconomics Group Bursting Analysis of Bifurcation diagram 4/27/2011 Chapter 10 – Neural Adaptation and Bursting Page 21 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 Comparison to 3-current model of spike rate adaptation 4/27/2011 Chapter 10 – Neural Adaptation and Bursting Page 25 Department of Economics / Computational Neuroeconomics Group Endogenous Bursting Neuron: in-vivo Difference to former model: • No stimulating current • Modulation back- and forth a saddle-node bifurcation 4/27/2011 Chapter 10 – Neural Adaptation and Bursting Page 26 Department of Economics / Computational Neuroeconomics Group Endogenous Bursting Neuron: in silico Simulation: PlantBurster.m X-C-projection of Phase space Time course of voltage V 4/27/2011 • 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) Chapter 10 – Neural Adaptation and Bursting Page 27 Department of Economics / Computational Neuroeconomics Group Wrap-up: Completing the single neuron taxonomy Class I (mammalian) Class II (squid/invertebrate) 4/27/2011 • Fast-spiking inhibitory neurons • Regular-spiking excitatory neurons • with spike rate adaptation • Current-driven bursting neurons • Chattering neurons • Fast-spiking neurons • Endogenous bursting neurons Chapter 10 – Neural Adaptation and Bursting Page 28 Department of Economics / Computational Neuroeconomics Group Separating limit cycles via intoxication VS Californian Aplysia (Seehase) Puffer Fish (Kugelfisch) Department of Economics / Computational Neuroeconomics Group Tetrodotoxin and Sushi • Tetrodotoxin (TTX) acts as nerve poison via blocking of the depolarizing Na+ channels • Neurons cannot create action potentials any longer 4/27/2011 Chapter 10 – Neural Adaptation and Bursting Page 30 Department of Economics / Computational Neuroeconomics Group Silencing all Na+-channels – in vivo 4/27/2011 Chapter 10 – Neural Adaptation and Bursting Page 31 Department of Economics / Computational Neuroeconomics Group Silencing all Na+-channels: in silico Without TTX With TTX • Still fluctuation due to X-C dynamics • No action potentials created 4/27/2011 Chapter 10 – Neural Adaptation and Bursting Page 32 Department of Economics / Computational Neuroeconomics Group Remaining limit cycle without Na+ current Simulation: PlantBursterTTX.m • X-C-projection of Phase space exhibits same limit cycle behavior • Modulation of X due to voltage changes vanish Without TTX 4/27/2011 Chapter 10 – Neural Adaptation and Bursting With TTX Page 33 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 Sensory cell bursting Mouse somatosensory cortex neuron 4/27/2011 Chapter 10 – Neural Adaptation and Bursting Cat visual cortex neuron Page 35 Department of Economics / Computational Neuroeconomics Group Driving Current 4/27/2011 Chapter 10 – Neural Adaptation and Bursting Page 36 Department of Economics / Computational Neuroeconomics Group Driving Current: differences to endogenous bursting model 4/27/2011 Chapter 10 – Neural Adaptation and Bursting Page 37 Department of Economics / Computational Neuroeconomics Group Driven bursting in a neocortical neuron Simulation: Chattering.m Time course of voltage V X-C-projection of phase space X-C Projection of Phase Space 0.4 0.35 20 10 0.3 0 I=0.2 -10 0.2 -20 X Potential (mV) 0.25 -30 0.15 -40 0.1 I=0.2 -50 0.05 -60 0 -70 -80 0 50 100 150 200 250 300 Time (ms) 350 400 450 -66 Potential (mV) -68 I=0.19 -70 -72 -74 -76 0 50 100 150 4/27/2011 200 250 300 Time (ms) 350 400 450 -0.05 -0.05 500 0 0.05 0.1 0.15 C 0.2 0.25 0.3 0.35 • 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 saddlenode bifurcation • One limit cycle driving the other to create bursts • But not autonomous due to V-dependence of X Chapter 10 – Neural Adaptation and Bursting Page 38 Department of Economics / Computational Neuroeconomics Group Wrap-up: Completing the single neuron taxonomy Class I (mammalian) Class II (squid/invertebrate) 4/27/2011 • Fast-spiking inhibitory neurons • Regular-spiking excitatory neurons • with spike rate adaptation • Current-driven bursting neurons • Chattering neurons • Fast-spiking neurons • Endogenous bursting neurons Chapter 10 – Neural Adaptation and Bursting Page 39 Department of Economics / Computational Neuroeconomics Group Dynamical Taxonomy of Class I neurons Fast-Spiking Inhibitory interneurons • Only 2 ion channel currents (Rinzel-model) • fast Na+ depolarization • slow K+ recovery • Constant spike rate: 1-400 Hz Regular Spiking Excitatory Neurons • Additional 3rd current • πΌπ΄π»π very slow after-hyperpolarizing K+ current • Enables spike rate adaptation Neocortical Bursting Cells 4/27/2011 • Additional 3rd & 4th current • πΌπ΄π»π very slow after-hyperpolarizing K+ current, mediated by Ca2+ concentration • πΌπ fast depolarizing Ca2+ current • Enables bursting, either intrinsic (πΌππ₯π‘ = 0) as pacemaker or driven by an external current Chapter 10 – Neural Adaptation and Bursting Page 40 Department of Economics / Computational Neuroeconomics Group Dynamical Taxonomy of Class I neurons Fast-Spiking Inhibitory interneurons ππ ππ+ − π πΎ+ + πΌ = −π11 π 2 ⋅ π − πππ 1 π π − π 12 ππ ππ₯π‘ ππ‘ ππ ππ‘ = 1 ππ −π + π21 π 2 Regular Spiking Excitatory Neurons ππ πΎ+ = β― − π13 1 π» π − πππ ππ‘ ππ» 1 = −π» + π31 π π − ππππ π‘ ππ‘ ππ» Neocortical Bursting Cells ππ πΆπ2+ = β― − π14 1 π π − πππ ππ‘ ππ» 1 = −π» + π31 1 π ππ‘ ππ» ππ 1 = −π + π31 π π − ππππ π‘ ππ‘ ππ 4/27/2011 Chapter 10 – Neural Adaptation and Bursting ππ =β― ππ‘ with ππ» β« ππ ππ =β― ππ‘ with ππ» β« ππ > ππ Page 41 Department of Economics / Computational Neuroeconomics Group 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 Chapter 10 – Neural Adaptation and Bursting Page 42 Department of Economics / Computational Neuroeconomics Group 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_DSC0125 7.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 Chapter 10 – Neural Adaptation and Bursting Page 43
