View
5
Download
0
Category
Preview:
Citation preview
Balance of Electric and Diffusion Forces
Ions flow into and out of the neuron under the forces of electricityand concentration gradients (diffusion).
The net result is a electric potential difference between the insideand outside of the cell — the membrane potential Vm.
This value represents an integration of the different forces, and anintegration of the inputs impinging on the neuron.
Electricity
Positive and negative charge (opposites attract, like repels).
Ions have net charge: Sodium (Na+), Chloride (Cl−), Potassium(K+), and Calcium (Ca++) (brain = mini ocean).
Current flows to even out distribution of positive and negativeions.
Disparity in charges produces potential (the potential to generatecurrent..)
Resistance
Ions encounter resistance when they move.Neurons have channels that limit flow of ions in/out of cell.
G
I
+ −
+
−V
The smaller the channel, the higher the resistance, the greater thepotential needed to generate given amount of current (Ohm’s law):
I =V
R(4)
Conductance G = 1/R, so I = V G
Diffusion
Constant motion causes mixing – evens out distribution.
Unlike electricity, diffusion acts on each ion separately — can’tcompensate one + ion for another..
(same deal with potentials, conductance, etc)
I = −DC (5)
(Fick’s First law)
Equilibrium
Balance between electricity and diffusion:
E = Equilibrium potential = amount of electrical potential neededto 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
InhibitorySynapticInput
ExcitatorySynapticInput
LeakCl−
Na+
Vm
Na/KPump
Vm
Vm
Vm
−70
+55
−70 K+
Cl−
Na+ K+
−70mV
0mV
Everything follows from the sodium pump, which creates the“dynamic tension” (compressing the spring, winding the clock) forsubsequent neural action.
The Neuron and its Ions
InhibitorySynapticInput
ExcitatorySynapticInput
LeakCl−
Na+
Vm
Na/KPump
Vm
Vm
Vm
−70
+55
−70 K+
Cl−
Na+ K+
−70mV
0mV
Glutamate → opens Na+ channels → Na+ enters (excitatory)
GABA → opens Cl- channels → Cl- enters if Vm ↑ (inhibitory)
Drugs and Ions
• Alcohol: closes Na
• General anesthesia: opens K
• Scorpion: opens Na and closes K
Putting it Together
Ic = gc(Vm − Ec) (7)
e = excitation (Na+)i = inhibition (Cl−)l = leak (K+).
Inet = ge(Vm − Ee) +
gi(Vm − Ei) +
gl(Vm − El) (8)
Vm(t + 1) = Vm(t) − dtvmInet (9)
or
Vm(t + 1) = Vm(t) + dtvmInet− (10)
Putting it Together: With Time
Ic = gc(t)gc(Vm(t) − Ec) (11)
e = excitation (Na+)i = inhibition (Cl−)l = leak (K+).
Inet = ge(t)ge(Vm(t) − Ee) +
gi(t)gi(Vm(t) − Ei) +
gl(t)gl(Vm(t) − El) (12)
Vm(t + 1) = Vm(t) − dtvmInet (13)
or
Vm(t + 1) = Vm(t) + dtvmInet− (14)
It’s Just a Leaky Bucket
Vm
ge
g i/l
excitation
inhibition/leak
ge = rate of flow into bucketgi/l = rate of “leak” out of bucketVm = balance between these forces
Or a Tug-of-War
Vm
excitation
inhibition
g i
ge
Vm
Ee
EiVm
Vm
In Action
−70− −65− −60− −55− −50− −45− −40− −35− −30−
V_m
I_net
05 10 15 20 25 30 35 40
−40−
−30−
−20−
−10−
0−
10−
20−
30−
40−
g_e = .2
g_e = .4
cycles
(Two excitatory inputs at time 10, of conductances .4 and .2)
Overall Equilibrium Potential
If you run Vm update equations with steady inputs, neuron settlesto new equilibrium potential.
To find, set Inet = 0, solve for Vm:
Vm =gegeEe + gigiEi + glglEl
gege + gigi + glgl(15)
Can now solve for the equilibrium potential as a function of inputs.
Simplify: ignore leak for moment, set Ee = 1 and Ei = 0:
Vm =gege
gege + gigi(16)
Membrane potential computes a balance (weighted average) ofexcitatory and inhibitory inputs.
Equilibrium Potential Illustrated
0.0 0.2 0.4 0.6 0.8 1.0g_e (excitatory net input)
0.0
0.2
0.4
0.6
0.8
1.0
V_m
Equilibrium V_m by g_e (g_l = .1)
Computational Neurons (Units) Summary
ge≈ i w ij > +
i
w ij
Vm= gege g gEe + i i Ei g+ lglEl
gege g+ igi + glgl
≈Vm−Θ[ ]+
Vm−Θ[ ]+ + 1j
net <x
x
βN=
yγγ
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 Na+ channels open if Vm > Θ, sharp rise in Vm.
Voltage Gated K+ channels open to reset spike.
−80−
−70−
−60−
−50−
−40−
−30−
−1−
−0.5−
0−
0.5−
1−
V_m
Rate CodeSpike
0 5 10 15 20 25 30 35 40
act
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?
Rate approximated by X-over-X-plus-1 ( xx+1):
yj =γ[Vm(t) − Θ]+
γ[Vm(t) − Θ]+ + 1(17)
which is like a sigmoidal function:
yj =1
1 + (γ[Vm(t) − Θ]+)−1(18)
compare to sigmoid: yj = 11+e
−ηj
γ is the gain: makes things sharper or duller.
Convolution with Noise
X-over-X-plus-1 has a very sharp threshold
Smooth by convolve with noise (just like “blurring” or“smoothing” in an image manip program):
Θ Vm
activity
Fit of Rate Code to Spikes
0−
10−
20−
30−
40−
50−
0−
0.2−
0.4−
0.6−
0.8−
1−
spike rate
V_m − Q−0.005 0 0.005 0.01 0.015
noisy x/x+1
Extra
Computing Excitatory Input Conductances
≈
A B
Σ β1N
ix ijw
Projections
1α
aa+b < ix ijw >
ix ijw
1α
ba+b
< ix ijw >
ge
s s
One projection per group (layer) of sending units.
Average weighted inputs 〈xiwij〉 = 1n
∑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) + dtvmInet− withbiological or normalized (0-1) parameters:
Parameter mV (0-1)Vrest -70 0.15El (K+) -70 0.15Ei (Cl−) -70 0.15Θ -55 0.25Ee (Na+) +55 1.00
Normalized used by default.
Detector vs. Computer
Computer DetectorMemory & Separate, Integrated,Processing general-purpose specialized
Operations Logic, arithmetic Detection(weighing & accumulatingevidence, evaluating,communicating)
Complex Arbitrary sequences Highly tuned sequencesProcessing of operations chained of detectors stacked
together in a program upon each other in layers
Recommended