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 3 4 1 1/23/2020 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 2 1/23/2020 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 9 Introducing systems and Levels of organization computational neuroscience • How groups of neurons work together to achieve intelligence • How the nervous system performs computations 12 13 3 1/23/2020 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? 14 15 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! 16 17 4 1/23/2020 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 18 19 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 20 21 5 1/23/2020 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 6 1/23/2020 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 26 27 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 7 1/23/2020 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 33 35 8 1/23/2020 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<vthresh 38 40 9 1/23/2020 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 10 1/23/2020 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”) 51 52 11 1/23/2020 Activation function An example sigmoid Function type g(2)= Linear Step g(1)= Threshold- linear g(0)= Sigmoid g(-4)= Radial-basis 53 54 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.