Balance of Electric and Diffusion Forces
Ions flow into and out of the neuron under the forces of electricity
and concentration gradients (diffusion).
The net result is a electric potential difference between the inside
and outside of the cell — the membrane potential Vm.
This value represents an integration of the different forces, and an
integration of the inputs impinging on the neuron.
Electricity
Positive and negative charge (opposites attract, like repels).
Ions have net charge: Sodium (N a+ ), Chloride (Cl−), Potassium
(K +), and Calcium (Ca++) (brain = mini ocean).
Current flows to even out distribution of positive and negative
ions.
Disparity in charges produces potential (the potential to generate
current..)
Resistance
Ions encounter resistance when they move.
Neurons have channels that limit flow of ions in/out of cell.
+
V
−
+
−
G
I
The smaller the channel, the higher the resistance, the greater the
potential needed to generate given amount of current (Ohm’s law):
I=
V
R
Conductance G = 1/R, so I = V G
(4)
Diffusion
Constant motion causes mixing – evens out distribution.
Unlike electricity, diffusion acts on each ion separately — can’t
compensate one + ion for another..
(same deal with potentials, conductance, etc)
I = −DC
(Fick’s First law)
(5)
Equilibrium
Balance between electricity and diffusion:
E = Equilibrium potential = amount of electrical potential needed
to counteract diffusion:
I = G(V − E)
(6)
Also:
Reversal potential (because current reverses on either side of E)
Driving potential (flow of ions drives potential toward this value)
The Neuron and its Ions
Inhibitory
Synaptic
Input
Cl−
−70
Leak
Excitatory Vm Cl−
Synaptic
Input
Na+
Na/K
Pump
+55
Vm
K+
−70
K+
Vm
Vm
Na+
−70mV
0mV
Everything follows from the sodium pump, which creates the
“dynamic tension” (compressing the spring, winding the clock) for
subsequent neural action.
The Neuron and its Ions
Inhibitory
Synaptic
Input
Cl−
−70
Leak
Excitatory Vm Cl−
Synaptic
Input
Na+
Na/K
Pump
+55
Vm
K+
−70
K+
Vm
Vm
Na+
−70mV
0mV
Glutamate → opens Na+ channels → Na+ enters (excitatory)
GABA → opens Cl- channels → Cl- enters if Vm ↑ (inhibitory)
Drugs and Ions
• Alcohol: closes N a
• General anesthesia: opens K
• Scorpion: opens N a and closes K
Putting it Together
Ic = gc(Vm − Ec)
(7)
Inet = ge(Vm − Ee) +
gi(Vm − Ei) +
gl (Vm − El )
(8)
Vm(t + 1) = Vm(t) − dtvmInet
(9)
Vm(t + 1) = Vm(t) + dtvmInet−
(10)
e = excitation (N a+ )
i = inhibition (Cl− )
l = leak (K +).
or
Putting it Together: With Time
Ic = gc(t)g¯c(Vm (t) − Ec)
(11)
e = excitation (N a+ )
i = inhibition (Cl− )
l = leak (K +).
Inet = ge(t)g¯e(Vm (t) − Ee) +
gi(t)g¯i(Vm(t) − Ei) +
gl (t)ḡl (Vm(t) − El )
(12)
Vm(t + 1) = Vm(t) − dtvmInet
(13)
Vm(t + 1) = Vm(t) + dtvmInet−
(14)
or
It’s Just a Leaky Bucket
excitation
g
e
Vm
inhibition/
leak
ge = rate of flow into bucket
gi/l = rate of “leak” out of bucket
Vm = balance between these forces
g
i/l
Or a Tug-of-War
excitation
ge
Ee
Vm
Vm
Vm
Vm
Ei
gi
inhibition
In Action
40− −30−
30− −35−
20− −40−
10− −45−
0− −50−
−10− −55−
−20−
−30−
−40−
I_net
g_e = .4
−60−
−65−
−70−
0
g_e = .2
V_m
5
10
15
20
cycles
25
30
35
40
(Two excitatory inputs at time 10, of conductances .4 and .2)
Overall Equilibrium Potential
If you run Vm update equations with steady inputs, neuron settles
to new equilibrium potential.
To find, set Inet = 0, solve for Vm:
Vm =
geg¯eEe + gig¯iEi + gl ḡl El
geg¯e + gig¯i + gl ḡl
(15)
Can now solve for the equilibrium potential as a function of inputs.
Simplify: ignore leak for moment, set Ee = 1 and Ei = 0:
geg¯e
Vm =
geg¯e + gig¯i
Membrane potential computes a balance (weighted average) of
excitatory and inhibitory inputs.
(16)
Equilibrium Potential Illustrated
Equilibrium V_m by g_e (g_l = .1)
1.0
V_m
0.8
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
g_e (excitatory net input)
1.0
Computational Neurons (Units) Summary
yj ≈
g g E +gg E +ggE
i i i
l l l
Vm = e e e
g g +gg +gg
e e i i
l l
γ [Vm− Θ]
+
γ [Vm− Θ] + 1
+
β
net = ge ≈ <x i w ij > +
N
w ij
xi
1. Weights = synaptic efficacy; weighted input = xiwij .
Net conductances (average across all inputs)
excitatory (net = ge(t)), inhibitory gi(t).
2. Integrate conductances using Vm update equation.
3. Compute output yj as spikes or rate code.
Thresholded Spike Outputs
Voltage gated N a+ channels open if Vm > Θ, sharp rise in Vm.
Voltage Gated K + channels open to reset spike.
−30−
1−
Rate Code
Spike
−40− 0.5−
−50−
0−
act
−60−
−70− −0.5−
−80−
−1−
0
V_m
5
10
15
20
25
30
35
40
In model: yj = 1 if Vm > Θ, then reset (also keep track of rate).
Rate Coded Output
Output is average firing rate value.
One unit = % spikes in population of neurons?
x ):
Rate approximated by X-over-X-plus-1 ( x+1
yj =
γ[Vm(t) − Θ]+
γ[Vm(t) − Θ]+ + 1
(17)
which is like a sigmoidal function:
1
yj =
1 + (γ[Vm(t) − Θ]+)−1
1
compare to sigmoid: yj =
−η
1+e j
γ is the gain: makes things sharper or duller.
(18)
Convolution with Noise
X-over-X-plus-1 has a very sharp threshold
activity
Smooth by convolve with noise (just like “blurring” or
“smoothing” in an image manip program):
Θ
Vm
Fit of Rate Code to Spikes
50− 1−
40− 0.8−
30− 0.6−
spike rate
noisy x/x+1
20− 0.4−
10− 0.2−
0− 0−
−0.005
0
0.005
V_m − Q
0.01
0.015
Extra
Computing Excitatory Input Conductances
g ≈Σ
e
1<
s a
xw >
i ij
a+b α
xw
i ij
1
β
N
b 1<
s
xw >
i ij
a+b α
xw
i ij
B
A
Projections
One projection per group (layer) of sending units.
!
1
Average weighted inputs $xiwij % = n i xiwij .
Bias weight β: constant input.
Factor out expected activation level α.
Other scaling factors a, s (assume set to 1).
Computing Vm
Use Vm(t + 1) = Vm(t) + dtvm Inet− with
biological or normalized (0-1) parameters:
Parameter
Vrest
El (K +)
Ei (Cl−)
Θ
Ee (N a+)
mV
-70
-70
-70
-55
+55
(0-1)
0.15
0.15
0.15
0.25
1.00
Normalized used by default.
Detector vs. Computer
Memory &
Processing
Computer
Separate,
general-purpose
Detector
Integrated,
specialized
Operations
Logic, arithmetic
Detection
(weighing & accumulating
evidence, evaluating,
communicating)
Complex
Processing
Arbitrary sequences
of operations chained
together in a program
Highly tuned sequences
of detectors stacked
upon each other in layers