Where
to
from
here?
Hodgkin-Huxley
Biophysical realism Simplified models
Ion channel physics Fundamental dynamics
Additional channels Analytical tractability
Geometry
Dendritic computation
Neurons
have
complicated
spatial
structures
Real Neurons Our Model
Geometry
matters!
Input
Input
Inject
current
at
the
cell
body
and
Inject
current
in
a
dendrite
and
record
effect
in
a
dendrite
record
effect
at
the
cell
body
Voltage decays with distance in passive membranes (How?)
Dayan and Abbott
Linear
cables
This
problem
sounds
familiar!
Lord Kelvin
Voltage
V
is
a
func8on
of
both
x
and
t
(1824-1907)
Developed cable
theory for
undersea cables
rm
and
ri
are
the
membrane
and
axial
resistances,
i.e.
the
resistances
of
a
thin
slice
of
the
cylinder
Johnston and Wu
The
cable
equation
Before
we
had:
Now
we
also
have
to
consider
ii,
the
current
down
the
cable,
due
to
voltage
changes
in
x.
That
current
works
against
internal
resistance,
ri
or
where Time constant
Space constant
How
does
voltage
decay
in
space?
⎛ | x| ⎞
⎜ − ⎟ Potential
decays
v( x) ∝ e ⎝ λ ⎠
exponentially
from
x
=
0
InEinite
Cable,
Constant
current
at
x
=
0
Dayan and Abbott
How
does
voltage
decay
over
space
and
time?
InEinite
Cable,
Current
pulse
at
t
=
0,
x
=
0
Potential
peaks
later
(and
at
lower
values)
for
points
further
away
from
input
General
solution:
Eilter
and
impulse
response
Diffusive
spread
Exponential
decay
OK:
now
what?
1.
The
geometry
can
be
extremely
complicated
Cable Equation
?
∂V 1 ∂ ⎛ 2 ∂V ⎞
cm = ⎜ a ⎟ − im + ie
∂t 2arL ∂x ⎝ ∂x ⎠
2.
And,
um,
ion
channels?
è
Quickly
becomes
intractable
to
solve
analytically
for
realistic
neurons
Solution:
Divide
and
Conquer
Compartmental
models
Decreasing
number
of
“compartments”
Each
compartment
=
one
dV/dt
equation
(usually
no
dependence
on
x)
The
gory
details
And now you
see why
Genesis and Coupling
NEURON were conductances
developed—
thank you!
Dayan and Abbott
What
do
dendrites
add
to
neuronal
computation?
London and Hausser, 2005
Delay
lines
in
sound
localization
Spain;
Scholarpedia
Direction
selectivity
V(t)
V(t)
Rall; London and Hausser, 2005
Enthusiastically
recommended
references
• Johnson
and
Wu,
Foundations
of
Cellular
Neurophysiology,
Chap
4
The
classic
textbook
of
biophysics
and
neurophysiology:
lots
of
problems
to
work
through.
Good
for
HH,
ion
channels,
cable
theory.
• Koch,
Biophysics
of
Computation
Insightful
compendium
of
ion
channel
contributions
to
neuronal
computation
• Izhikevich,
Dynamical
Systems
in
Neuroscience
An
excellent
primer
on
dynamical
systems
theory,
applied
to
neuronal
models
• Magee,
Dendritic
integration
of
excitatory
synaptic
input,
Nature
Reviews
Neuroscience,
2000
Review
of
interesting
issues
in
dendritic
integration
• London
and
Hausser,
Dendritic
Computation,
Annual
Reviews
in
Neuroscience,
2005
Review
of
the
possible
computational
space
of
dendritic
processing
