Balance of Electric and Diffusion Forces

Transcription

1 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 V m. This value represents an integration of the different forces, and an integration of the inputs impinging on the neuron.

2 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 negative ions. Disparity in charges produces potential (the potential to generate current..)

3 Resistance Ions encounter resistance when they move. Neurons have channels that limit flow of ions in/out of cell. + V + I G 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

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) (Fick s First law) I = DC (5)

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)

6 The Neuron and its Ions Na+ Na/K Pump Excitatory Synaptic Input Vm Cl +55 Na+ Inhibitory Synaptic Input Vm Vm 70 Cl K+ 70mV 0mV 70 Leak Vm K+ Everything follows from the sodium pump, which creates the dynamic tension (compressing the spring, winding the clock) for subsequent neural action.

7 The Neuron and its Ions Na+ Na/K Pump Excitatory Synaptic Input Vm Cl +55 Na+ Inhibitory Synaptic Input Vm Vm 70 Cl K+ 70mV 0mV 70 Leak Vm K+ Glutamate opens Na+ channels Na+ enters (excitatory) GABA opens Cl- channels Cl- enters if V m (inhibitory)

8 Drugs and Ions Alcohol: closes Na General anesthesia: opens K Scorpion: opens Na and closes K

9 Putting it Together I c = g c (V m E c ) (7) e = excitation (Na + ) i = inhibition (Cl ) l = leak (K + ). or I net = g e (V m E e ) + g i (V m E i ) + g l (V m E l ) (8) V m (t + 1) = V m (t) dt vm I net (9) V m (t + 1) = V m (t) + dt vm I net (10)

10 Putting it Together: With Time e = excitation (Na + ) i = inhibition (Cl ) l = leak (K + ). or I c = g c (t)ḡ c (V m (t) E c ) (11) I net = g e (t)ḡ e (V m (t) E e ) + g i (t)ḡ i (V m (t) E i ) + g l (t)ḡ l (V m (t) E l ) (12) V m (t + 1) = V m (t) dt vm I net (13) V m (t + 1) = V m (t) + dt vm I net (14)

11 It s Just a Leaky Bucket excitation g e V m inhibition/ leak g i/l g e = rate of flow into bucket g i/l = rate of leak out of bucket V m = balance between these forces

12 Or a Tug-of-War excitation g e E e V m V m V m E i V m g i inhibition

13 I_net V_m In Action g_e =.4 g_e = cycles (Two excitatory inputs at time 10, of conductances.4 and.2)

14 Overall Equilibrium Potential If you run V m update equations with steady inputs, neuron settles to new equilibrium potential. To find, set I net = 0, solve for V m : V m = g eḡ e E e + g i ḡ i E i + g l ḡ l E l g e ḡ e + g i ḡ i + g l ḡ l (15) Can now solve for the equilibrium potential as a function of inputs. Simplify: ignore leak for moment, set E e = 1 and E i = 0: V m = g eḡ e g e ḡ e + g i ḡ i (16) Membrane potential computes a balance (weighted average) of excitatory and inhibitory inputs.

15 Equilibrium Potential Illustrated 1.0 Equilibrium V_m by g_e (g_l =.1) 0.8 V_m g_e (excitatory net input)

16 Computational Neurons (Units) Summary V m = g g g g e e E e + i i E g i + g l l E l g g + g g + g g e e i i l l w ij y j γ γ [ V m Θ] + V m Θ [ ] net g e <x i w ij > + β = N x i 1. Weights = synaptic efficacy; weighted input = x i w ij. Net conductances (average across all inputs) excitatory (net = g e (t)), inhibitory g i (t). 2. Integrate conductances using V m update equation. 3. Compute output y j as spikes or rate code.

17 Thresholded Spike Outputs Voltage gated Na + channels open if V m > Θ, sharp rise in V m. Voltage Gated K + channels open to reset spike. 1 Rate Code 30 Spike act V_m In model: y j = 1 if V m > Θ, then reset (also keep track of rate).

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 ): y j = γ[v m(t) Θ] + γ[v m (t) Θ] which is like a sigmoidal function: y j = compare to sigmoid: y j = 1 1+e η j γ is the gain: makes things sharper or duller. (17) (γ[v m (t) Θ] + ) 1 (18)

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): activity Θ Vm

20 Fit of Rate Code to Spikes spike rate noisy x/x V_m Q

21 Extra

22 Computing Excitatory Input Conductances s a a+b Σ g e 1 < α x w i ij > x w i ij 1 N β b 1 s a+b α x w i ij < x i w ij > A B Projections One projection per group (layer) of sending units. Average weighted inputs x i w ij = 1 n Bias weight β: constant input. Factor out expected activation level α. Other scaling factors a, s (assume set to 1). i x i w ij.

23 Computing V m Use V m (t + 1) = V m (t) + dt vm I net with biological or normalized (0-1) parameters: Parameter mv (0-1) V rest E l (K + ) E i (Cl ) Θ E e (Na + ) Normalized used by default.

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 of operations chained of detectors stacked together in a program upon each other in layers