Department of Economics / Computational Neuroeconomics Group Neural Adaptation and Bursting or: A dynamical taxonomy of neurons April 27 th, 2011 Lars Kasper

Department of Economics / Computational Neuroeconomics Group Introduction and Link to last sessions

Department of Economics / Computational Neuroeconomics Group Symbols & Numbers Vmembrane potential Rrecovery variable (related to K + ) Hconductance variable (related to slow K + current, I AHP ) Cvery slow K + (I AHP ) conductance mediated by intracellular Ca 2+ concentration XCa 2+ conductance, rapid depolarizing current I A rapid transient K + current I AHP slow afterhyperpolarizing K + current I ADP slow afterdepolarizing current (fast R and slow X comb.) +55, +48 mVNa + equilibrium potential +140 mVCa 2+ equilibrium potential -95, -92 mV K + equilibrium potential -70, mVResting membrane potential 4/27/2011Chapter 10 – Neural Adaptation and BurstingPage 3

Department of Economics / Computational Neuroeconomics Group Overview of Introduced Neuron Models ModelHodgin-Huxley/RinzelConnor et al. Rose&Hindmarsh Neuron typeClass II (squid axon)Class I (fast-spiking, inhib. cortical neuron) Experimental phenomena explained High frequency firing ( Hz) High and low frequency firing ( Hz) Included Ion CurrentsDepolarizing 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 4/27/2011Chapter 10 – Neural Adaptation and BurstingPage 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 Ca 2+ ) This dynamic net current fluctuation will lead to complex behavior due to recurring back- and forth-crossings of bifurcation boundaries 4/27/2011Chapter 10 – Neural Adaptation and BurstingPage 5

Department of Economics / Computational Neuroeconomics Group 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) 4/27/2011Chapter 10 – Neural Adaptation and BurstingPage 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 Ca 2+ 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 4/27/2011Chapter 10 – Neural Adaptation and BurstingPage 10

Department of Economics / Computational Neuroeconomics Group After-hyperpolarization via slow K + current 4/27/2011Chapter 10 – Neural Adaptation and BurstingPage 11

Department of Economics / Computational Neuroeconomics Group Explanation via reduction of effective driving current 4/27/2011Chapter 10 – Neural Adaptation and BurstingPage 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 Capability of the model 4/27/2011Chapter 10 – Neural Adaptation and BurstingPage 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: I AHP can be blocked or reduced by neuromodulators (ACh, histamine, norepinephrine, serotonin)

Department of Economics / Computational Neuroeconomics Group 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) 4/27/2011Chapter 10 – Neural Adaptation and BurstingPage 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 I AHP due to K + current:

Department of Economics / Computational Neuroeconomics Group Bursting Neurons 4/27/2011Chapter 10 – Neural Adaptation and BurstingPage 17 Simulation: HHburster.m with I=0.14, 0.18

Department of Economics / Computational Neuroeconomics Group Bursting Neurons 4/27/2011Chapter 10 – Neural Adaptation and BurstingPage 18 Simulation: HHburster.m with I=0.14, 0.18 V-R projection of phase space trajectories (red)

Department of Economics / Computational Neuroeconomics Group Bursting analysis of bifurcation diagram 4/27/2011Chapter 10 – Neural Adaptation and BurstingPage 19

Department of Economics / Computational Neuroeconomics Group Bursting analysis of bifurcation diagram 4/27/2011Chapter 10 – Neural Adaptation and BurstingPage 20 I net ↑ H ↑ V ↑ V ↓ I net ↓ H ↓ Action potential Action potential AP vanishes AP vanishes

Department of Economics / Computational Neuroeconomics Group Bursting Analysis of Bifurcation diagram 4/27/2011Chapter 10 – Neural Adaptation and BurstingPage 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 Ca 2+ -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 Ca 2+ channels) C is Ca 2+ -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 4/27/2011Chapter 10 – Neural Adaptation and BurstingPage 25

Department of Economics / Computational Neuroeconomics Group Endogenous Bursting Neuron: in-vivo 4/27/2011Chapter 10 – Neural Adaptation and BurstingPage 26 Difference to former model: No stimulating current Modulation back- and forth a saddle-node bifurcation

Department of Economics / Computational Neuroeconomics Group Endogenous Bursting Neuron: in silico 4/27/2011Chapter 10 – Neural Adaptation and BurstingPage 27 Simulation: PlantBurster.m X-C-projection of Phase 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 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) 4/27/2011Chapter 10 – Neural Adaptation and BurstingPage 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 Tetrodotoxin and Sushi 4/27/2011Chapter 10 – Neural Adaptation and BurstingPage 30 Tetrodotoxin (TTX) acts as nerve poison via blocking of the depolarizing Na + channels Neurons cannot create action potentials any longer

Department of Economics / Computational Neuroeconomics Group Silencing all Na + -channels – in vivo 4/27/2011Chapter 10 – Neural Adaptation and BurstingPage 31

Department of Economics / Computational Neuroeconomics Group Silencing all Na + -channels: in silico 4/27/2011Chapter 10 – Neural Adaptation and BurstingPage 32 Without TTX Still fluctuation due to X-C dynamics No action potentials created With TTX

Department of Economics / Computational Neuroeconomics Group Remaining limit cycle without Na + current 4/27/2011Chapter 10 – Neural Adaptation and BurstingPage 33 Simulation: PlantBursterTTX.m Without TTXWith 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 Ca 2+ -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 4/27/2011Chapter 10 – Neural Adaptation and BurstingPage 35 Mouse somatosensory cortex neuron Cat visual cortex neuron

Department of Economics / Computational Neuroeconomics Group Driving Current 4/27/2011Chapter 10 – Neural Adaptation and BurstingPage 36

Department of Economics / Computational Neuroeconomics Group Driving Current: differences to endogenous bursting model 4/27/2011Chapter 10 – Neural Adaptation and BurstingPage 37

Department of Economics / Computational Neuroeconomics Group Driven bursting in a neocortical neuron 4/27/2011Chapter 10 – Neural Adaptation and BurstingPage 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 I=0.19 I=0.2 X-C-projection of phase spaceTime course of voltage V

Department of Economics / Computational Neuroeconomics Group 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) 4/27/2011Chapter 10 – Neural Adaptation and BurstingPage 39

Department of Economics / Computational Neuroeconomics Group Dynamical Taxonomy of Class I neurons 4/27/2011Chapter 10 – Neural Adaptation and BurstingPage 40 Fast-Spiking Inhibitory interneurons Fast-Spiking Inhibitory interneurons Regular Spiking Excitatory Neurons Regular Spiking Excitatory Neurons Only 2 ion channel currents (Rinzel-model) fast Na + depolarization slow K + recovery Constant spike rate: Hz Neocortical Bursting Cells Neocortical Bursting Cells

Department of Economics / Computational Neuroeconomics Group Dynamical Taxonomy of Class I neurons 4/27/2011Chapter 10 – Neural Adaptation and BurstingPage 41 Fast-Spiking Inhibitory interneurons Fast-Spiking Inhibitory interneurons Regular Spiking Excitatory Neurons Regular Spiking Excitatory Neurons Neocortical Bursting Cells Neocortical Bursting Cells

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 Ca 2+ ) This dynamic net current fluctuation leads to complex behavior due to recurring back- and forth-crossings of bifurcation boundaries 4/27/2011Chapter 10 – Neural Adaptation and BurstingPage 42

Department of Economics / Computational Neuroeconomics Group Picture Sources /1000px-Tetrodotoxin.svg.png 7.JPG => Chapter 9 and 10 4/27/2011Chapter 10 – Neural Adaptation and BurstingPage 43