Cable equation solution

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CS 621 Artificial IntelligenceLecture 34 -
08/11/05Guest Lecture byProf. Rohit
ManchandaBiological Neurons - II
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The human brain
Seat of consciousness and cognition Perhaps the
most complex information processing machine in
nature Historically, considered as a monolithic
information processing machine
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Beginners Brain Map
Forebrain (Cerebral Cortex) Language, maths,
sensation, movement, cognition, emotion
Midbrain Information Routing involuntary
controls
Cerebellum Motor Control
Hindbrain Control of breathing, heartbeat, blood
circulation
Spinal cord Reflexes, information highways
between body brain
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Brain a computational machine?

• Information processing brains vs computers
• - brains better at perception / cognition
• - slower at numerical calculations
• Evolutionarily, brain has developed algorithms
  most suitable for survival
• Algorithms unknown the search is on
• Brain astonishing in the amount of information it
  processes
• Typical computers 109 operations/sec
• Housefly brain 1011 operations/sec

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Brain facts figures

• Basic building block of nervous system nerve
  cell (neuron)
• 1012 neurons in brain
• 1015 connections between them
• Connections made at synapses
• The speed events on millisecond scale in
  neurons, nanosecond scale in silicon chips

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Neuron - classical

• Dendrites
• Receiving stations of neurons
• Don't generate action potentials
• Cell body
• Site at which information received is
• integrated
• Axon
• Generate and relay action potential
• Terminal
• Relays information to next neuron
• in the pathway

http//www.educarer.com/images/brain-nerve-axon.jp
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Membrane Biophysics OverviewPart 1 Resting
membrane potential
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Resting Membrane Potential

• Measurement of potential between ICF and ECF
• Vm Vi - Vo
• ICF and ECF at isopotential separately.
• ECF and ICF are different from each other.

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Resting Membrane Potential - recording

• Electrode wires can not be inserted in the cells
  without damaging them (cell membrane thickness
  7nm)
• Solution Glass microelectrodes (Tip diameter 10
  nm)
• Glass ? Non conductor
• Therefore, while pulling a capillary after
  heating, it is filled with KCl and tip of
  electrode is open and KCl is interfaced with a
  wire.

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R.m.p. - towards a theory

• Ionic concentration gradients across biological
  cell membrane

Mammalian muscle (rmp -75 mV) Mammalian muscle (rmp -75 mV) Mammalian muscle (rmp -75 mV)
ECF ICF
Cations Cations Cations
Na 145 mM 12 mM
K 4 mM 155 mM
Anions Anions Anions
Cl- 120 mM 4 mM
Frog muscle (rmp -80 mV) Frog muscle (rmp -80 mV) Frog muscle (rmp -80 mV)
ECF ICF
Cations Cations Cations
Na 109 mM 4 mM
K 2.2 mM 124 mM
Anions Anions Anions
Cl- 77 mM 1.5 mM
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R.m.p. - towards a theory

• Ionic concentration gradients squid axon (rmp
  -60 mV)

• Ionic concentration ratios

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Ionic concentration ratios across biological cell
membranes
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Trans-membrane Ionic Distributions
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Resting potential as a K equilibrium (Nernst)
potential
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Resting Membrane Potential Nernst Eqn
Consider values for typical concentration
ratios EK -90 mV ENa 60 mV r.m.p. -60
to 80 mV
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Goldman-Hodgkin-Katz (GHK) eqn
Taking values of R,T F and dividing throughout
by PK
Consider ? V. large, v. small, and intermediate
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Equivalent Circuit Model Resting Membrane
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Equivalent Circuit Model including Na pump
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Membrane Biophysics OverviewPart 2 Action
potential
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ACTION POTENTIAL
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ACTION POTENTIAL Ionic mechanisms
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Action Potential Na and K Conductance
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Action Potential Propagation

• Non decremental, constant velocity

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Hippocampus location
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Hippocampal Network Connections
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Membrane Biophysics OverviewPart 3 Synaptic
transmission potentials
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Canonical neurons Neuroscience
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Synapses Chemical Electrical
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Transmission Chemical Electrical
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Chemical Transmission
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Postsynaptic Electrical Effects
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Synaptic Integration The Canonical Picture
Action potential Output signal
Axon Output line
Action potential
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The Perceptron Model A perceptron is a
computing element with input lines having
associated weights and the cell having a
threshold value. The perceptron model is
motivated by the biological neuron.
Output y
Threshold ?
w1
wn
Wn-1
x1
Xn-1
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• Features of Perceptron
• Input output behavior is discontinuous and the
  derivative does not exist at Swixi ?
• Swixi - ? is the net input denoted as net
• Referred to as a linear threshold element -
  linearity because of x appearing with power 1
• y f(net) Relation between y and net is
  non-linear

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• Perceptron / ANN neuron
• Neurophysiological basis of
• Input Signs
• Input Weights

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Dendrites Synapses in Real Life !
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Neuron morphologies
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In the Retina