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