[PPT] - Conductance-Based Models Fundamentals of Computational Neuroscience, PowerPoint Presentation

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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

E-mail: btzhang@bi.snu.ac.kr This material is available online at http://bi.snu.ac.kr/

Fundamentals of Computational Neuroscience, T. P . Trappenberg, 2010.

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Outline

2.1 2.2 2.3 2.4 2.5 2.6 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

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2.1 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

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2.2 Neurons are specialized cells

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2.2.1 Structural properties (1)

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Fig. 2.1

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2.2.1 Structural properties (2)

 Morphologies of different

neurons

 Pyramidal cell from the motor cortex (B)  Granule neuron from olfactory bulb (C)  Spiny neuron from the caudate nucleus (D)  Golgi-stained Purkinje cell from the cerebellum (E)

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Fig. 2.1

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2.2.2 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

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2.2.3 Membrane potential



Membrane potential  The difference between the electric potential within the cell and its surrounding

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m

V

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2.2.4 Ion channels (1)



The permeability of the membrane to certain ions is achieved by ion channels

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Fig. 2.2

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2.2.4 Ion channels (2)

 Major ion channels

 Pump: use energy  Channel: use difference of ions concentration

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2.2.4 Ion channels (3)

 Resting potential

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mV Vrest 65  

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Supplement: Equilibrium potential and Nernst equation

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2.3 Basic synaptic mechanisms

 Signal transduction within the cell is mediated by electrical

potentials.

 Electrical synapse or gap-junctions  Chemical synapse

 Synaptic plasticity

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Fig. 2.3

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2.3.1 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)

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2.3.2 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

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 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

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peak

t t NMDA non m

e t w V

/  

   

2 1

/ /

) (

  t t m NMDA m

e e V c V

 

    

Fig. 2.4

peak

t t NMDA non m

e t w V

/  

    

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2.3.4 Superposition of PSP

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 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

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2.4 The generation of action potentials

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 Spike or action potential (AP)

 Voltage-dependent sodium channel

Start rising phase

 Neurotransmitter-gated ion channels

Depolarize

 Voltage-dependent sodium channels

Influx of Na+

 Falling phase

 Hyperpolarization  The sodium channels inactive  Potassium channels open

Fig. 2.5
Fig. 2.6

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2.4.3 Hodgkin-Huxley equations (1)

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 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

) (

ion ion ion

E V g I  

ion

E

ion

g

V

h m g g n g g

Na Na K K 3 4

 

Fig. 2.7

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2.4.3 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

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dx dt = - 1 t x(V) [x - x0(V)]

Fig. 2.8

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2.4.3 Hodgkin-Huxley equations (3)

 Hodgkin-Huxley model

 C, capacitance  I(t), external current

 Three ionic currents

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

  

ion ion

t I I dt dV C ) (

)] ( [ )] ( [ )] ( [ ) ( ) ( ) ( ) (

3 4

V h h dt dh V m m dt dm V n n dt dn t I E V g E V h m g E V n g dt dV C

h m n L L Na Na K K

                   

Fig. 2.7

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2.4.4 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)

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Fig. 2.9

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2.4.5 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

f neurons even further

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2.4.6 Propagation of action potentials

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2.5 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

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Fig. 2.10

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2.5.1 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,

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) , ( 4 ) , ( ) , ( ) , (

2 2 2

t x I R d V t x V t t x V x t x V

inj m rest m m m

             

i m

R dR 4  

) , ( t x Vm 

m

R

i

R

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2.5.1 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

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m m m

C R   

m

R

m

C

x x m

e v e v x V

 

   

 2 1

) (

rest x rest m

V e V V V    



) (

) ), , ( ( 4 ) , ( ) , ( ) , (

2 2 2

t t x V I R d V t x V t t x V x t x V

m inj m rest m m m

             

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2.5.2 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

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2 1 1 2 2

) ( ) ( ) ( 2 ) , (

j i j j j j m

x x t V t V V x t x V      

  

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2.5.3 Neuron simulators

 Ex) Neuron, Genesis

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Fig. 2.11

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2.6 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

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)] ( [ ) ( ) )( ( ) (

0 V

R R dt dR t I E V V g E V R g dt dV C

R Na Na K K

         

m

 V

R

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2.6 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.

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)] ( 3 [ )] ( [ )] ( [ ) ( ) ( ) ( ) ( ) )( ( V T H dt dH V T T dt dT V R R dt dR t I E V H g E V T g E V R g E V V g dt dV C

H T R h H T T k k Na Na

                     

2 4 2 4 2 4

810 116 . 205 . 4 ) ( 210 . 3 037 . 24 . 1 ) ( 810 . 33 476 . 8 . 17 ) ( V V V T V V V R V V V gNa

  

        

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2.6 Above and beyond the Hodgkin-Huxley neuron: fatigue, bursting, simplifications (3)

 Simulation of the Wilson model

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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

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