SLIDE 1
Balance of Electric and Diffusion Forces Ions flow into and out of - - PowerPoint PPT Presentation
Balance of Electric and Diffusion Forces Ions flow into and out of - - PowerPoint PPT Presentation
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
SLIDE 2
SLIDE 3
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 the potential needed to generate given amount of current (Ohm’s law): I = V R (4) Conductance G = 1/R, so I = V G
SLIDE 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 (5) (Fick’s First law)
SLIDE 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)
SLIDE 6
The Neuron and its Ions
Inhibitory Synaptic Input Excitatory Synaptic Input Leak
Cl− Na+
Vm Na/K Pump 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) for subsequent neural action.
SLIDE 7
The Neuron and its Ions
Inhibitory Synaptic Input Excitatory Synaptic Input Leak
Cl− Na+
Vm Na/K Pump 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)
SLIDE 8
Drugs and Ions
- Alcohol: closes Na
- General anesthesia: opens K
- Scorpion: opens Na and closes K
SLIDE 9
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)
- r
Vm(t + 1) = Vm(t) + dtvmInet− (10)
SLIDE 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)
- r
Vm(t + 1) = Vm(t) + dtvmInet− (14)
SLIDE 11
It’s Just a Leaky Bucket
Vm ge gi/l excitation inhibition/ leak
ge = rate of flow into bucket gi/l = rate of “leak” out of bucket Vm = balance between these forces
SLIDE 12
Or a Tug-of-War
V m excitation inhibition gi ge V m Ee Ei V m V m
SLIDE 13
In Action
−70− −65− −60− −55− −50− −45− −40− −35− −30− V_m I_net 5 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)
SLIDE 14
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 = ge ¯ geEe + gi¯ giEi + gl¯ glEl ge ¯ ge + gi¯ gi + gl¯ gl (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 = ge ¯ ge ge ¯ ge + gi¯ gi (16) Membrane potential computes a balance (weighted average) of excitatory and inhibitory inputs.
SLIDE 15
Equilibrium Potential Illustrated
0.0 0.2 0.4 0.6 0.8 1.0 g_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)
SLIDE 16
Computational Neurons (Units) Summary
ge≈ i wij > + i wij Vm = gege g g Ee + i i Ei g + lglEl gege g + igi + glgl
≈
Vm−Θ
[ ]+
Vm−Θ
[ ]+ + 1
j net <x x
β
N = y γ γ
- 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.
SLIDE 17
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 Code Spike 5 10 15 20 25 30 35 40 act
In model: yj = 1 if Vm > Θ, then reset (also keep track of rate).
SLIDE 18
Rate Coded Output
Output is average firing rate value. One unit = % spikes in population of neurons? Rate approximated by X-over-X-plus-1 (
x x+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 =
1 1+e−ηj
γ is the gain: makes things sharper or duller.
SLIDE 19
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
SLIDE 20
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 0.005 0.01 0.015 noisy x/x+1
SLIDE 21
Extra
SLIDE 22
Computing Excitatory Input Conductances
≈ A B
Σ
β
1 N i x ij w Projections 1
α
a a+b
<
i x ij w > i x ij w 1
α
b a+b
<
i x ij w > ge s s
One projection per group (layer) of sending units. Average weighted inputs xiwij = 1
n
- i xiwij.
Bias weight β: constant input. Factor out expected activation level α. Other scaling factors a, s (assume set to 1).
SLIDE 23
Computing Vm
Use Vm(t + 1) = Vm(t) + dtvmInet− with biological or normalized (0-1) parameters: Parameter mV (0-1) Vrest
- 70
0.15 El (K+)
- 70
0.15 Ei (Cl−)
- 70
0.15 Θ
- 55
0.25 Ee (Na+) +55 1.00 Normalized used by default.
SLIDE 24
Detector vs. Computer
Computer Detector Memory & Separate, Integrated, Processing general-purpose specialized Operations Logic, arithmetic Detection (weighing & accumulating evidence, evaluating, communicating) Complex Arbitrary sequences Highly tuned sequences Processing
- f operations chained
- f detectors stacked