1 2. Neurons and Conductance-Based Models Lecture Notes on Brain and Computation Byoung-Tak Zhang Biointelligence Laboratory School of Computer Science and Engineering Graduate Programs in Cognitive Science, Brain Science and Bioinformatics Brain-Mind-Behavior Concentration Program Seoul National University This material is available online at Fundamentals of Computational Neuroscience, T. P. Trappenberg, 2002.

(C) 2009 SNU CSE Biointelligence Lab(C) 2009 SNU CSE Biointelligence Lab, 2 Outline Modeling biological neurons Neurons are specialized cells Basic synaptic mechanisms The generation of action potentials: Hodgkin-Huxley equations Dendritic trees, the propagation of action potentials, and compartmental models Above and beyond the Hodgkin-Huxley neuron: fatigue, bursting, and simplifications

(C) 2009 SNU CSE Biointelligence Lab(C) 2009 SNU CSE Biointelligence Lab, Modeling biological neurons The networks of neuron-like elements  The heart of many information-processing abilities of brain The working of single neurons  Information transmission Simplified versions of the real neurons  Make computations with large numbers of neurons tractable  Enable certain emergent properties in networks  Nodes The sophisticated computational abilities of neurons The computational approaches used to describe single neurons 3

(C) 2009 SNU CSE Biointelligence Lab(C) 2009 SNU CSE Biointelligence Lab, Neurons are specialized cells 4

(C) 2009 SNU CSE Biointelligence Lab(C) 2009 SNU CSE Biointelligence Lab, Structural properties (1) 5 Fig. 2.1 (A) Schematic neuron that is similar in appearance to pyramidal cells in the neocortex. The components outlined in the drawing are, however, typical for major neuron types.

(C) 2009 SNU CSE Biointelligence Lab(C) 2009 SNU CSE Biointelligence Lab, Structural properties (2) Fig. 2.1 (B-E) Examples of morphologies of different neurons.  (B) Pyramidal cell from the motor cortex  (C) Granule neuron from olfactory bulb  (D) Spiny neuron from the caudate nucleus  (E) Golgi-stained Purkinje cell from the cerebellum. 6

(C) 2009 SNU CSE Biointelligence Lab(C) 2009 SNU CSE Biointelligence Lab, Information-processing mechanisms Neurons can receive signals from many other neurons Synapses (contact site)  Presynaptic (from axon)  Postsynaptic (to dendrite or cell body) Signal = action potential  Electronic pulse 7

(C) 2009 SNU CSE Biointelligence Lab(C) 2009 SNU CSE Biointelligence Lab, Membrane potential Membrane potential  The difference between the electric potential within the cell and its surrounding 8

(C) 2009 SNU CSE Biointelligence Lab(C) 2009 SNU CSE Biointelligence Lab, Ion channels (1) The permeability of the membrane to certain ions is achieved by ion channels 9 Fig. 2.2 Schematic illustrations of different types of ion channels. (A) Leakage channels are always open. (B) Neurotransmitter-gated ion channel that opens when a neurotransmitter molecule binds to the channel protein, which in turn changes the shape of the channel protein so that specific ions can pass through. (C) The opening of voltage- gated ion channels depends on the membrane potential. This is indicated by a little wire inside the neurons and a grounding wire outside the neuron. Such ion channels can in addition be neurotransmitter-gated (not shown in this figure). (D) Ion pumps are ion channels that transport ions against the concentration gradients. (E) Some neurotransmitters bind to receptor molecules which triggers a whole cascade of chemical reactions in neurons which produce secondary messengers which in turn can influence ion channels

(C) 2009 SNU CSE Biointelligence Lab(C) 2009 SNU CSE Biointelligence Lab, Ion channels (2) Major ion channels  Pump: use energy  Channel: use difference of ions concentration 10

(C) 2009 SNU CSE Biointelligence Lab(C) 2009 SNU CSE Biointelligence Lab, Ion channels (3) Resting potential 11

(C) 2009 SNU CSE Biointelligence Lab(C) 2009 SNU CSE Biointelligence Lab, Supplement Equilibrium potential and Nernst equation 12

(C) 2009 SNU CSE Biointelligence Lab(C) 2009 SNU CSE Biointelligence Lab, Basic synaptic mechanisms Signal transduction within the cell is mediated by electrical potentials.  Electrical synapse or gap-junctions  Chemical synapse Synaptic plasticity Chemical synapses and neurotransmitters neurotransmitters stored in synaptic vesicles  glutamate (Glu)  gamma-aminobutyric acid (GABA)  Dopamine (DA)  acetylcholine (ACh) synaptic cleft (a small gap of only a few micrometers) Receptor (channel) and Postsynaptic potential (PSP) Fig. 2.3 (A) Schematic illustration of chemical synapses and (B) an electron microscope photo of a synaptic terminal.

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(C) 2009 SNU CSE Biointelligence Lab(C) 2009 SNU CSE Biointelligence Lab, Excitatory and inhibitory synapses Different types of neurotransmitters Excitatory synapse  PSP: depolarization  Neurotransmitters trigger the increase of the membrane potential  Neurotransmitter: Glu, ACh Inhibitory synapse  PSP: hyperpolarization  Neurotransmitters trigger the decrease of the membrane potential  Neurotransmitter: GABA Non-standard synapses  Influence ion channels in an indirect way  Modulation 16

(C) 2009 SNU CSE Biointelligence Lab(C) 2009 SNU CSE Biointelligence Lab, The time-course of postsynaptic potentials Excitatory postsynaptic potential (EPSP) resulting from non-NMDA receptors w: amplitude factor  strength of EPSP or efficiency of the synapse f(t)=t·exp(-t): α-function  functional form of a PSP Inhibitory postsynaptic potential (IPSP) resulting from non-NMDA receptors EPSP resulting from NMDA receptor 17 (2.1) (2.2) Fig. 2.4 Examples of an EPSP (solid line) and IPSP (dashed line) as modeled by an α-function after the synaptic delay. The dotted lines represent two examples of the difference of two exponential functions with different amplitudes. Note that the time scale is variable and not meant to fit experimental data in the illustration. Indeed, NMDA synapses are often much slower than non-NMDA synapses, often showing their maximal value long after the peak in the effect of the non-NMDA synapses.

(C) 2009 SNU CSE Biointelligence Lab(C) 2009 SNU CSE Biointelligence Lab, Superposition of PSP 18 Electrical potentials have the physical property  They superimpose as the sum of individual potentials. Linear superposition of synaptic input Nonlinear voltage-current relationship Nonlinear interaction  Divisive  Shunting inhibition

(C) 2009 SNU CSE Biointelligence Lab(C) 2009 SNU CSE Biointelligence Lab, The generation of action potentials 19 Fig. 2.6 Schematic illustration o the minimal mechanisms necessary for the generation of a spike. The resting potential of a cell is maintained by a leakage channel through which potassium ions can flow as a result of concentration differences between the inside of the cell and the surrounding fluid. A voltage-gated sodium channel allows the influx of positively charged sodium ions and thereby the depolarization of the cell. After a short time the sodium channel is blocked and a voltage-gated potassium channel opens. This results in a hyperpolarization of the cell. Finally, the hyperpolarization causes the inactivation of the voltage-gated channels and a return to the resting potential. Fig. 2.5 Typical form of an action potential, redrawn from an oscilloscope picture of Hodgkin and Huxley.

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(C) 2009 SNU CSE Biointelligence Lab(C) 2009 SNU CSE Biointelligence Lab, Hodgkin-Huxley equations (1) 21 Quantified the process of spike generation Input Current  Electric conductance  Membrane potential relative to the resting potential  Equilibrium potential K+, Na+ conductance dependent  n, The activation of the K channel  m, The activation of the Na channel  h, The inactivation of the Na channel (2.3) (2.4) (2.5) Fig. 2.7 A Circuit representation of the Hodgkin-Huxley model. This circuit includes a capacitor on which the membrane potential can b e measured and three resistors together with their own battery modeling the ion channels; two are voltage-dependent and one is static.

(C) 2009 SNU CSE Biointelligence Lab(C) 2009 SNU CSE Biointelligence Lab, Hodgkin-Huxley equations (2) n, m, h have the same form of first-order differential-equation x should be substituted by each of the variables n, m and h 22 (2.6) Fig. 2.8 (A) The equilibrium functions and (B) time constants for the three variables n, m, and h in the Hodgkin-Huxley model with parameters used to model the gain axon of the squid.

(C) 2009 SNU CSE Biointelligence Lab(C) 2009 SNU CSE Biointelligence Lab, Hodgkin-Huxley equations (3) Hodgkin-Huxley model  C, capacitance  I(t), external current Three ionic currents 23 (2.7) (2.8) (2.9) (2.10) Fig. 2.7 A Circuit representation of the Hodgkin-Huxley model. This circuit includes a capacitor on which the membrane potential can b e measured and three resistors together with their own battery modeling the ion channels; two are voltage- dependent and one is static.

(C) 2009 SNU CSE Biointelligence Lab(C) 2009 SNU CSE Biointelligence Lab, Numerical integration A. A Hodgkin-Huxley neuron responds with constant firing to a constant external current. B. The dependence of the firing rate with the external current (nonlinear curve). (dashed line: noise added) 24 Fig. 2.9 (A) A Hodgkin-Huxley neuron responds with constant firing to a constant external current of I ext = 10. (B) The dependence of the firing rate with the strength of the external current shows a sharp onset of firing around I c ext = 6 (solid line). High-frequency noise in the current has the tendency to ‘linearize’ these response characteristics (dashed line).

(C) 2009 SNU CSE Biointelligence Lab(C) 2009 SNU CSE Biointelligence Lab, Refractory period Absolute refractory period  The inactivation of the sodium channel makes it impossible to initiate another action potential for about 1ms.  Limiting the firing rates of neurons to a maximum of about 1000Hz Relative refractory period  Due to the hyperpolarizing action potential it is relatively difficult to initiate another spike during this time period.  Reduced the firing frequency of neurons even further 25

(C) 2009 SNU CSE Biointelligence Lab(C) 2009 SNU CSE Biointelligence Lab, Propagation of action potentials 26

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(C) 2009 SNU CSE Biointelligence Lab(C) 2009 SNU CSE Biointelligence Lab, Dendritic trees, the propagation of action potentials, and compartmental models Axons with active membranes able to generate action potential But, dendirtes are a bit more like passive conductors  The long cable  Cable theory Compartmental models 28 Fig Short cylindrical compartments describing small equipotential pieces of a passive dendrite or small pieces of a active dendrite or axon when including the necessary ion channels. (A) Chains and (B) branches determine the boundary conditions of each compartment. (C) The topology of single neurons can be reconstructed with compartments, and such models can be simulated using numerical integration techniques.

(C) 2009 SNU CSE Biointelligence Lab(C) 2009 SNU CSE Biointelligence Lab, Cable theory (1) Cable equation : the spatial-temporal variation of an electric potential The potential of the cable at each location of the cable, the physical properties of the cable and has the dimensions of Ωcm,  Ex) cylindrical cable of diameter d,  The specific resistance of the membrane,  The specific intracellular resistance of the cable, 29 (2.11) (2.12)

(C) 2009 SNU CSE Biointelligence Lab(C) 2009 SNU CSE Biointelligence Lab, Cable theory (2) The time constant  The resistance of the membrane,  The capacitance, the capacitance per unit area, Stable configuration Semi-infinite cable Nonlinear cable equation, include voltage-dependent ion channels as in the Hodgkin-Huxley equation 30 (2.13) (2.14) (2.15) (2.16)

(C) 2009 SNU CSE Biointelligence Lab(C) 2009 SNU CSE Biointelligence Lab, Physical shape of neurons The physical shape of neurons  Simple homogeneous linear cable is to divide the cable into small pieces, compartment.  Each compartment is governed by a cable equation for a finite cable  The boundary conditions 31 (2.17)

(C) 2009 SNU CSE Biointelligence Lab(C) 2009 SNU CSE Biointelligence Lab, Neuron simulators Ex) Neuron, Genesis 32 Fig Example of the NEURON simulation environment. The example shows a simulation with a reconstructed pyramidal cell, in which a short current pulse was injected at t = 1 ms at the location of the dendrite indicated by the dot and the cursor in the window on the left.

(C) 2009 SNU CSE Biointelligence Lab(C) 2009 SNU CSE Biointelligence Lab, Above and beyond the Hodgkin-Huxley neuron: fatigue, bursting, simplifications (1) Simplification of the Hodgkin-Huxley model  The time constant is so small for all values of.  The rate of inactivation of the Na+ channel is approximately reciprocal to the opening of the K+ channel.  Neocortical neurons often show no inactivation of the fast Na+ channel.  The recovery of the membrane potential 33 (2.18) (2.19)

(C) 2009 SNU CSE Biointelligence Lab(C) 2009 SNU CSE Biointelligence Lab, Above and beyond the Hodgkin-Huxley neuron: fatigue, bursting, simplifications (2) Extensions of the Hodgkin-Huxley model  Ca2+ channel, a dynamic gating variable T.  Slow hyperpolarizing current, Ca2+-mediated K+ channel, a dynamic gating variable H. 34 (2.20) (2.21) (2.22) (2.23) (2.24) (2.25) (2.26)

(C) 2009 SNU CSE Biointelligence Lab(C) 2009 SNU CSE Biointelligence Lab, Above and beyond the Hodgkin-Huxley neuron: fatigue, bursting, simplifications (3) Simulation of the Wilson model 35 Fig Simulated spike train of the Wilson model. The upper graph simulates fast spiking neurons (FS) typical of inhibitory neurons in the mammalian neocortex (τ R = 1.5ms, g T = 0.25, g H = 0). The middle graph models regular spiking neurons (RS) with longer spikes (τ R = 4.2ms, g T = 0.1). The slow calcium-mediated potassium channel (g H = 5) is responsible for the slow adaptation in the spike frequency. The lower graph demonstrate that even more complex behavior, typical of mammalian neocortical neurons, is incorporated in the Wilson model. The parameters (τ R = 4.2ms, g T = 2.25, g H = 9.5) result I a typical bursting behavior, including a typical after- depolarizing potential (ADP).

(C) 2009 SNU CSE Biointelligence Lab(C) 2009 SNU CSE Biointelligence Lab, Fig Simulated spike train of the Wilson model. The upper graph simulates fast spiking neurons (FS) typical of inhibitory neurons in the mammalian neocortex (τ R = 1.5ms, g T = 0.25, g H = 0). The middle graph models regular spiking neurons (RS) with longer spikes (τ R = 4.2ms, g T = 0.1). The slow calcium-mediated potassium channel (g H = 5) is responsible for the slow adaptation in the spike frequency. The lower graph demonstrate that even more complex behavior, typical of mammalian neocortical neurons, is incorporated in the Wilson model. The parameters (τ R = 4.2ms, g T = 2.25, g H = 9.5) result I a typical bursting behavior, including a typical after-depolarizing potential (ADP). 36

(C) 2009 SNU CSE Biointelligence Lab(C) 2009 SNU CSE Biointelligence Lab, Conclusion Neurons utilize a variety of specialized biochemical mechanisms for information processing transmission  Membrane potential, ion channel  Action potential, neurotransmitter  Propagation, refractory period Conductance-based models  Hodgkin-Huxley equation Compartmental models  Cable theory Neuron simulators  Neuron, Genesis 37