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.
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..)
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
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)
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 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.
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)
Drugs and Ions Alcohol: closes Na General anesthesia: opens K Scorpion: opens Na and closes K
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)
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)
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
Or a Tug-of-War excitation g e E e V m V m V m E i V m g i inhibition
40 30 20 10 0 10 20 30 40 30 35 40 45 50 55 60 65 70 0 I_net V_m In Action g_e =.4 g_e =.2 5 10 15 20 25 30 35 40 cycles (Two excitatory inputs at time 10, of conductances.4 and.2)
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.
Equilibrium Potential Illustrated 1.0 Equilibrium V_m by g_e (g_l =.1) 0.8 V_m 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 g_e (excitatory net input)
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 [ ] + + 1 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.
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 40 0.5 50 0 60 70 0.5 act V_m 80 1 0 5 10 15 20 25 30 35 40 In model: y j = 1 if V m > 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 ( x x+1 ): y j = γ[v m(t) Θ] + γ[v m (t) Θ] + + 1 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) 1 1 + (γ[v m (t) Θ] + ) 1 (18)
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
Fit of Rate Code to Spikes 50 1 40 0.8 30 0.6 20 0.4 spike rate noisy x/x+1 10 0.2 0 0 0.005 0 0.005 0.01 0.015 V_m Q
Extra
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.
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 -70 0.15 E l (K + ) -70 0.15 E i (Cl ) -70 0.15 Θ -55 0.25 E e (Na + ) +55 1.00 Normalized used by default.
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