计算神经科学
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An Introduction to the Mathematical Modeling of Primary Visual Cortex
Louis Tao 陶乐天
北京大学 生命科学学院 生物信息中心 Center for Bioinformatics, College of Life Sciences Peking University
© Copyright 2010 Center for Bioinformatics, Peking University
Systems Neuroscience
1. Neuronal activity and behavior (correlations) 2. Coding: action potential frequency (rate coding) action potential timing (temporal coding) population coding (e.g., V1) attractor/transient coding (e.g., olfaction) 3. Modeling: neuronal network as dynamical system 4. Today’s example: visual cortex
The Neural Signal 信号
~ 100 mV
动作电位 Action Potentials: a sequence of short, electrical pulses in the membrane potential of a neuron • V ~ 100 mv in 1-2 msec • Action potential propagates at constant speed along axons – velocity ~ 1 cm in 1 msec • Action potential does not change shape as it propagates • Stereotyped shape: each action potential has the same shape and form therefore the form of the action potential does not carry information • Spike Trains: Sequence of action potential can be regular or irregular
Voltage Clamp
Delcomyn (1997)
Device to measure the current necessary to keep the desired V
© Copyright 2010 Center for Bioinformatics, Peking University
Collaborators David Cai 蔡申瓯 (SJTU, CIMS NYU), Gregor Kovacic (RPI), Dave McLaughlin (CIMS NYU), Adi Rangan (CIMS NYU), Robert Shapley (CNS NYU), Michael Shelley (CIMS NYU)
Feature detection (visual systems, olfactory system, …) Coincidence / timing (auditory system) Memory (hippocampus) Sensory-motor (eye saccades, …) Neural Code: firing rate, spike timing
104 cells & 1 km wiring in 1 mm3 of cortex
© Copyright 2010 Center for Bioinformatics, Peking University
Neurons 神经元
• Information processing units 信息处理加工单元 • 1010-1013 neurons in mammalian brains 哺乳动物大脑 • 104 cell bodies and roughly 1 km of ‘wiring’ per mm3 • Different shapes, sizes, functions, … 不同形状、大小、功能 • Spiking vs. Analog neurons 锋电位（动作电位） (“analog neurons,” e.g., bipolar and amacrine cells in retina, sensory-motor neurons of invertebrates, … 视网膜里的双极细胞与无长突细胞，无脊椎动物的感觉-运动 神经元，等) • Many other cells (e.g., glia cells 胶质细胞) in cortex: to supply energy, to provide structural stability, …, and not directly involved in information processing
© Copyright 2010 Center for Bioinformatics, Peking University
Computational Neuroscience
• • •
– – – – –
What “computations” are performed by neurons & neuronal networks? How are these computations done? What?
ISI: Inter-Spike Interval • Information carried by spike trains
© Copyright 2010 Center for Bioinformatics, Peking University
Action Potentials 动作电位
• Action potential (spikes): voltage pulses responding to ‘strong’ input • All-or-None electrical events, initiated near the cell body, propagates along axons (at roughly constant velocity and amplitude) • Hodgkin & Huxley (1950s) studied various ionic currents individually (using pharmacological blocks); used squid giant axon (1/2 mm diameter) • Action Potential generation involves 2 major, voltagedependent currents: sodium and potassium; individual ionic current obeys Ohm’s law
© Copyright 2010 Center for Bioinformatics, Peking University
Neuronal and Network Dynamics
• Individual neurons
– Hodgkin-Huxley equation – Action potential generation (ionic channel, …) – Synaptic currents
m Cm / g Leak 10 msec
I syn Gsyn t Vm Vsyn
© Copyright 2010 Center for Bioinformatics, Peking University
Hodgkin Huxley Neuron Model
Vm d 2Vm I Na I K I Leak I syn I inj Cm R x 2 t
• Network Activity
– Wilson-Cowan type models – Neuronal networks: integrate-and-fire neuronal networks, HH neuronal networks, …
© Copyright 2010 Center for Bioinformatics, Peking University
I ion Gion V Vion Gsyn 1 Rion Vion RT cion ,inside ln zF cion ,outside
© Copyright 2010 Center for Bioinformatics, Peking University
Hodgkin Huxley Neuron Model
Vm d 2Vm I Na I K I Leak I syn I inj Cm R x 2 t
voltage step command Vrest
+25mV
g K g K n4
gK IK V EK
Hodgkin Huxley Neuron Model
Vm d 2Vm I Na I K I Leak I syn I inj Cm R x 2 t
Squid Giant Axon
1963 Nobel Prize
© Copyright 2010 Center for Bioinformatics, Peking University
VNa 55 mV, g Na 115 mS/cm 2 VK 90 mV, g K 36 mS/cm 2 VLeak 65 mV, g Leak 0.1 mS/cm 2 , Cm 1 F/cm 2
I leak Gleak Vm Vrest
Synaptic current: induced by action potentials of other neurons
•
– –
How?
Cell level: molecular and biophysical Network & systems level
•
– – –
What & How to Study?
Cellular: membrane potential, ion channels, synaptic mechanisms Extracellular: firing rates, spike times, statistics of spike trains, … Systems: fMRI, optical imaging, …
交大致远学院 计算神经科学 夏季短期班 24 May 2010
© Copyright 2010 Center for Bioinformatics, Peking University
Neuronal Networks Are Cwenku.baidu.commplex
~1011 neurons & 1015 connections
n
dn n n dt
Estimate n and n
from time course
© Copyright 2010 Center for Bioinformatics, Peking University
Hodgkin Huxley Neuron Model
Vm d 2Vm I Na I K I Leak I syn I inj Cm R x 2 t
I Na g Na m3 h V VNa I K g K n 4 V VK
Leak current:
Louis Tao 陶乐天
北京大学 生命科学学院 生物信息中心 Center for Bioinformatics, College of Life Sciences Peking University
© Copyright 2010 Center for Bioinformatics, Peking University
Systems Neuroscience
1. Neuronal activity and behavior (correlations) 2. Coding: action potential frequency (rate coding) action potential timing (temporal coding) population coding (e.g., V1) attractor/transient coding (e.g., olfaction) 3. Modeling: neuronal network as dynamical system 4. Today’s example: visual cortex
The Neural Signal 信号
~ 100 mV
动作电位 Action Potentials: a sequence of short, electrical pulses in the membrane potential of a neuron • V ~ 100 mv in 1-2 msec • Action potential propagates at constant speed along axons – velocity ~ 1 cm in 1 msec • Action potential does not change shape as it propagates • Stereotyped shape: each action potential has the same shape and form therefore the form of the action potential does not carry information • Spike Trains: Sequence of action potential can be regular or irregular
Voltage Clamp
Delcomyn (1997)
Device to measure the current necessary to keep the desired V
© Copyright 2010 Center for Bioinformatics, Peking University
Collaborators David Cai 蔡申瓯 (SJTU, CIMS NYU), Gregor Kovacic (RPI), Dave McLaughlin (CIMS NYU), Adi Rangan (CIMS NYU), Robert Shapley (CNS NYU), Michael Shelley (CIMS NYU)
Feature detection (visual systems, olfactory system, …) Coincidence / timing (auditory system) Memory (hippocampus) Sensory-motor (eye saccades, …) Neural Code: firing rate, spike timing
104 cells & 1 km wiring in 1 mm3 of cortex
© Copyright 2010 Center for Bioinformatics, Peking University
Neurons 神经元
• Information processing units 信息处理加工单元 • 1010-1013 neurons in mammalian brains 哺乳动物大脑 • 104 cell bodies and roughly 1 km of ‘wiring’ per mm3 • Different shapes, sizes, functions, … 不同形状、大小、功能 • Spiking vs. Analog neurons 锋电位（动作电位） (“analog neurons,” e.g., bipolar and amacrine cells in retina, sensory-motor neurons of invertebrates, … 视网膜里的双极细胞与无长突细胞，无脊椎动物的感觉-运动 神经元，等) • Many other cells (e.g., glia cells 胶质细胞) in cortex: to supply energy, to provide structural stability, …, and not directly involved in information processing
© Copyright 2010 Center for Bioinformatics, Peking University
Computational Neuroscience
• • •
– – – – –
What “computations” are performed by neurons & neuronal networks? How are these computations done? What?
ISI: Inter-Spike Interval • Information carried by spike trains
© Copyright 2010 Center for Bioinformatics, Peking University
Action Potentials 动作电位
• Action potential (spikes): voltage pulses responding to ‘strong’ input • All-or-None electrical events, initiated near the cell body, propagates along axons (at roughly constant velocity and amplitude) • Hodgkin & Huxley (1950s) studied various ionic currents individually (using pharmacological blocks); used squid giant axon (1/2 mm diameter) • Action Potential generation involves 2 major, voltagedependent currents: sodium and potassium; individual ionic current obeys Ohm’s law
© Copyright 2010 Center for Bioinformatics, Peking University
Neuronal and Network Dynamics
• Individual neurons
– Hodgkin-Huxley equation – Action potential generation (ionic channel, …) – Synaptic currents
m Cm / g Leak 10 msec
I syn Gsyn t Vm Vsyn
© Copyright 2010 Center for Bioinformatics, Peking University
Hodgkin Huxley Neuron Model
Vm d 2Vm I Na I K I Leak I syn I inj Cm R x 2 t
• Network Activity
– Wilson-Cowan type models – Neuronal networks: integrate-and-fire neuronal networks, HH neuronal networks, …
© Copyright 2010 Center for Bioinformatics, Peking University
I ion Gion V Vion Gsyn 1 Rion Vion RT cion ,inside ln zF cion ,outside
© Copyright 2010 Center for Bioinformatics, Peking University
Hodgkin Huxley Neuron Model
Vm d 2Vm I Na I K I Leak I syn I inj Cm R x 2 t
voltage step command Vrest
+25mV
g K g K n4
gK IK V EK
Hodgkin Huxley Neuron Model
Vm d 2Vm I Na I K I Leak I syn I inj Cm R x 2 t
Squid Giant Axon
1963 Nobel Prize
© Copyright 2010 Center for Bioinformatics, Peking University
VNa 55 mV, g Na 115 mS/cm 2 VK 90 mV, g K 36 mS/cm 2 VLeak 65 mV, g Leak 0.1 mS/cm 2 , Cm 1 F/cm 2
I leak Gleak Vm Vrest
Synaptic current: induced by action potentials of other neurons
•
– –
How?
Cell level: molecular and biophysical Network & systems level
•
– – –
What & How to Study?
Cellular: membrane potential, ion channels, synaptic mechanisms Extracellular: firing rates, spike times, statistics of spike trains, … Systems: fMRI, optical imaging, …
交大致远学院 计算神经科学 夏季短期班 24 May 2010
© Copyright 2010 Center for Bioinformatics, Peking University
Neuronal Networks Are Cwenku.baidu.commplex
~1011 neurons & 1015 connections
n
dn n n dt
Estimate n and n
from time course
© Copyright 2010 Center for Bioinformatics, Peking University
Hodgkin Huxley Neuron Model
Vm d 2Vm I Na I K I Leak I syn I inj Cm R x 2 t
I Na g Na m3 h V VNa I K g K n 4 V VK
Leak current: