CISC 3250 Systems Neuroscience

1/23/2020

CISC 3250 Systems (and Computational) Neuroscience Systems Neuroscience • How the nervous system performs computations

• How groups of neurons work together to achieve intelligence

• Requirement for the Integrative Neuroscience Professor Daniel Leeds major [email protected] • Elective in Computer and Information Science

JMH 332 2

Objectives Recommended student background Prerequisite: To understand information processing in • Officially: CISC 1800/1810 Intro to Programming biological neural systems from computational or CISC 2500 Information and Data and anatomical perspectives Management • Understand the function of key components of the nervous system Computer • Understand how to make mathematical Math models of cognition science • Understand how to use computational tools to Some calculus Some programming examine neural data

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Textbook(s) Website

Fundamentals of Computational Neuroscience, Second Edition, http://storm.cis.fordham.edu/leeds/cisc3250/ by Trappenberg • Suggested Go online for • We will focus on the ideas and study – Announcements a relatively small set of equations – Lecture slides – Course materials/handouts – Assignments Computational Cognitive Neuroscience, by O’Reilly et al.

• Optional, alternate perspective 5 6

Requirements Matlab • Attendance and participation – 1 unexcused absence allowed Popular tool in scientific computing for: – Ask and answer questions in class • Finding patterns in data • Homework: Roughly 5 across the semester • Plotting results • Exams • Running simulations – 1 midterm and 1 final – 2 shorter quizzes Student license for $50 on Mathworks site • Don’t cheat Available in computers at JMH 302 and – You may discuss course topics with other LL 612 students, but you must answer homeworks

yourself (and exams!) yourself 7 8

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Your instructor Prof. Leeds’ Projects in Computational Neuroscience Prof. Daniel Leeds E-mail: [email protected] • Computer vision models for Office hours: Mon 12-1, Thurs 2-3 cortical vision Office: JMH 332

Memor y apple • Effects of head trauma on car bear cortical cognition

Introducing systems and Levels of organization computational neuroscience

• How groups of neurons work together to achieve intelligence

• How the nervous system performs computations

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From a psychological perspective… Systems neuroscience

Regions of the central nervous system associated with particular elements of cognition

What are elements • Visual object recognition of cognition?

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Systems neuroscience Computational neuroscience

Regions of the central nervous system Strategy used by the nervous system to solve associated with particular elements of cognition problems

• Visual object recognition • Visual object perception • Motion planning and execution through biological • Learning and remembering hierarchical model “HMAX” – Show pictures!

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Computational neuroscience as Marr’s three levels for “HMAX” vision “theory of the brain” • Computational theory: Goal is to recognize David Marr’s three levels of analysis (1982): objects • Computational theory: What is the • Representation and algorithm: computational goal and the strategy to achieve – Input: Pixels of light and color it? – Output: Label of object identity • Representation and algorithm: What are the – Conversion: Through combining local visual input and output for the computation, and how properties do you mathematically convert input to output? • Hardware implementation: • Hardware implementation: How do the physical components perform the computation? – Visual properties “computed” by networks of firing neurons in object recognition pathway

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Levels of organization Course outline

• Philosophy of neural modeling • The neuron – biology and input/output behavior • Learning in the neuron • Neural systems and neuroanatomy • Representations in the brain • Memory/learning • Motor control Plus: Matlab • Perception programming

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The neuron Neuron membrane voltage • Building block of all the systems we will study • Cell with special properties • Voltage difference across cell membrane – Soma (cell body) can have 5-100 μm diameter, but – Resting potential: ~-65 mV axon can stretch over 10-1000 cm in length – Action potential: quick upward spike in voltage – Receives input from neurons through dendrites – Sends output to neurons through axon

potential (mV) potential time (ms)

22 Example neural signals 23

The action potential Inter-neuron communication

• Action potential begins at axon hillock and Neuron receives input from 1000s of other neurons travels down axon • Excitatory input can increase spiking • At each axon terminal, • Inhibitory input can decrease spiking spike results in release A synapse links neuron A with neuron B of neurotransmitters • Neuron A is pre-synaptic: • Neurotransmitters axon terminal outputs NTs (NTs) attach to • Neuron B is post-synaptic: dendrite of another dendrite takes NTs as input neuron, causing voltage change in this second neuron 24 25

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More on neuron membrane voltage Patch clamp experiment

• Given no input, membrane stays at resting • Attach electrode to neuron potential (~ -65 mV) • Raise/drop voltage on electrode Inputs: • Measure nearby voltage (with • Excitation temporarily increases potential another electrode)

• Inhibition temporarily decreases potential input

Simplification of Continual drive to remain at rest neurophysiology

experiment nearby

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More on the action potential Modeling voltage over time Equations focusing on change in voltage v 1. Accumulated excitation passes certain level Components: • Resting state potential (voltage) EL 2. Non-linear increase in membrane voltage • Input voltages RI • Time t 푑푣(푡) 3. Rapid reset 휏 = − 푣 푡 − 퐸 + 푅퐼(푡) 푑푡 퐿 change towards incorporate new resting state input information

http://commons.wikimedia.org/wiki/File:Action_potential.svg 28 29 CC User: Chris 73

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Simulation Applying dv/dt step-by-step

EL=-65mV v(0ms)=-65mV 휏=1 RI(t)=20mV (from t=0ms to 1000ms) • Initial voltage time step: 10ms 푑푣(푡) 휏 = − 푣 푡 − 퐸 + 푅퐼(푡) • Time interval for update 푑푡 퐿 푑푣(0ms) 10 • v(10ms) = v(0ms) + x • Input at each time 푑푡 1000 10 = -65 + [-(-65- -65) + 20] x 10 1000 = -65 + 20 x 1000 • Apply rule to compute new voltage at each = -64.8 푑푣(10ms) 10 time • v(20ms) = v(10ms) + x 푑푡 1000 10 = -64.8 + [-(-64.8- -65) + 20] x 10 1000 = -64.8 + -0.2+20 x 101000 = -64.8 + 19.8 x 1000 30 = -64.602 32

Applying dv/dt step-by-step Changing model terms EL=-65mV v(0ms)=-65mV 휏=1 RI(t)=20mV (from t=0ms to 1000ms) 휏 푑푣(푡) has inverse effect 휏 = − 푣 푡 − 퐸 + 푅퐼(푡) time step: 10ms 푑푡 퐿 • increase 휏 decreases update speed • decrease 휏 increases update speed 푑푣(0ms) 10 • v(30ms) = v(20ms) + x 푑푡 1000 10 = -64.602 + [-(-64.602- -65) + 20] x RI(t) has linear effect 1000 10 = -64.602 + 19.602 x • increase RI(t) increases update speed 1000 = -64.40598 • decrease RI(t) decreases update speed

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Voltage over time: reset Voltage over time 푑푣(푡) ↑ 휏 = − 푣 푡 − 퐸 + 푅퐼(푡) 10 푑푣(푡) 푑푡 퐿 휏 = − 푣 푡 − 퐸 + 푅퐼(푡) 푑푡 퐿 When voltage passes threshold vthresh, voltage reset to vres -45 f v(t )=vthresh f -50 v(t +δ)=vres Example:

δ is small positive number close vthresh=-42mV -55 to 0 vreset =-65mV -60 v(120ms)=-45mV -65 v(130ms)=-43mV 0 10 20 30 40 50 60 70 80 90 100 v(140ms)=-41.5mV Simulated Biological v(150ms)=-65mV36 37

Below and above threshold 0 Accumulating information over inputs -10 -20 -30 -40 -50 -60 -70 0 100 200 300 400 500 0 100 200 300 400 500 Positive and negative weighted inputs from +15mv input +50mv input dendrites wα added together: EL=-65mV Newly added: 푅퐼 푡 = 푤푗훼푗(푡) 푗 If input constant for long time RI(t)= k mV j is index over dendrites; first-pass model Output v(t) will plateau to EL+k if EL+k

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A Accumulating inputs A Accumulating inputs

-40 D1 -50 D1 -50 A -60 A -70 -60 D2 -80 D2

-70 -90 0 200 400 600 800 1000 0 200 400 600 800 1000 w =1 w =1 훼1(t) 1 훼1 푡 1 +20 +20 w =1 w =-3 0 2 0 2 훼2(t) 훼2 푡 10 10 0 0 41 43

v(t) Chemical level: NT receptors -50 D1 -60 A -70 Pre-synaptic: 훼 • Amount of NT released -80 D2 Post-synaptic: w -90 0 200 400 600 800 1000 • Number of receptors in dendrite membrane 훼1 푡 w1=1 +20 w1 [0 0 20 20 20 … 20 20 …] • Efficiency of receptors + w2 [0 0 0 0 0 … 10 10 …] +w or –w 0 w2=-3 • 훼2 푡 10 1x [0 0 20 20 20 … 20 20 …] Reflect excitation or inhibition 0 + 3x [0 0 0 0 0 … 10 10 …] • One NT type per synapse [0 0 20 20 20 … 20 20 …] • Fixed sign per NT +20 푅퐼 푡 + [0 0 0 0 0 … -30 -30 …] 0 [0 0 20 20 20 … -10 -1044 …] 48 -10

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Form of dendrite input 푑푣(푡) 푅퐼 푡 = 푤푗훼푗 (푡) 휏 = − 푣 푡 − 퐸퐿 + 푅퐼(푡) 푑푡 푗 Pre-synaptic neuron spikes -50

Neurotransmitter (NT) released -55 New pre-synaptic NT received by post-synaptic inputs at f dendrite at time t -60 • 34 ms • 68 ms Post-synaptic voltage rises and • 100 ms then fades, α(t) -65 • 135 ms α(t) -70 푅퐼 푡 = 푤푗훼푗 (푡) 0 20 40 60 80 100 120 140 160 푗 f t t 49 50

“Leaky integrate-and-fire” neuron Activation function

• Sum inputs from Often non-linear relation between dendrite input 푅퐼 푡 = 푤푗훼푗(푡) and axon output 푑푣(푡) dendrites (“integral”) 휏 = − 푣 푡 − 퐸 + 푔(푅퐼 푡 ) 푗 푑푡 퐿

• Decrease voltage 푅퐼 푡 = 푤푗훼푗(푡) Sum inputs 푑푣(푡) 푗 towards resting state 휏 = − 푣 푡 − 퐸 + 푅퐼(푡) 푑푡 퐿 (“leak”)

푔(푅퐼 푡 ) Apply (non-linear?) • Reset after passing 푓 transformation to input 푣 푡 + 훿 = 푣푟푒푠 threshold (“fire”)

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Activation function An example sigmoid

Function type g(2)= Linear

Step g(1)=

Threshold- linear g(0)=

Sigmoid g(-4)= Radial-basis

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Tuning curves Variations in activation functions Some single neurons fire in response to “perceiving” a quality in the world • Activation function has fixed shape – Sigmoid is S shape, Radial is Bell shape

• By default, transition between 0 and 1

• Some details of shape may vary – Smallest and highest value Adrian, Henry et al., – Location of transition between values J Physiol 1926. J Neurophys

1974. 56 57

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Neural coding Time coding at t=290ms Perception, action, and other cognitive states 1 represented by firing of neurons • Coding by rate: high rate of pre-synaptic 2 spiking causes post-synaptic spiking • Coding by spike timing: multiple pre-synaptic 3 neurons spiking together causes post-synaptic spiking 4

0 100 200 300 400ms Neuron index Neuron

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Rate coding: 3.5 – 5.5s Spike time coding, ???s

0 1s 2s 3s 4s 5s 6s 7s 8s 60 0 1s 2s 3s 4s 5s 6s 7s 8s 61

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Inhibition can be informative Computing spike rate

Inputs of interest can produce • Add spikes over a period of time • Below-normal spike rate 푛푢푚 푠푝푖푘푒푠 푖푛 Δ푇 • Decreased synchrony among neurons 푣 푡 = Δ푇 Coding through rate inhibition, roughly in 2-3s interval • Average spikes over a set of neurons Take note of baseline. Rate and time coding 1 푛푢푚 푠푝푖푘푒푠 푖푛 푁 푛푒푢푟표푛푠 are deviations from 퐴 푡 = lim Δ푇→0 Δ푇 푁 0s 1s 2s 3s 4s 5s 6s baseline 63 64