Neuron. Detector Model. Understanding Neural Components in Detector Model. Detector vs. Computer. Detector. Neuron. output. axon

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1 Neuron Detector Model 1 The detector model. 2 Biological properties of the neuron. 3 The computational unit. Each neuron is detecting some set of conditions (e.g., smoke detector). Representation is what is detected. Neurons feed on each other s outputs layers of ever more complicated detectors. (Things can get very complex in terms of content, but each neuron is still carrying out basic detector function). output integration inputs Detector 1 / 29 2 / 29 Detector vs. Computer Understanding Neural Components in Detector Model 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 output integration inputs Detector axon cell body, membrane potential dendrites synapses Neuron 3 / 29 4 / 29

2 A Real Neuron Basic Properties of a Neuron It s a cell: body, membrane, nucleus, DNA, RNA, proteins, etc. Membrane has channels, passing ions (salt water). Cell has electrical potential (voltage), integrated in cell body, activates action potential output in axon, releases neurotransmitter. Neurotransmitter activates potential via dendritic synaptic input channels. Excitation and inhibition are transmitted by different neurons! 5 / 29 6 / 29 The Synapse Weight = Synaptic Efficacy Dendrite Receptors Spine Synaptic efficacy = activity of presynaptic (sending) neuron communicated to postsynaptic (receiving) neuron: Neuro transmitter (glutamate) Vesicles mglu AMPA NMDA Na+ Ca++ Ca++ Terminal Button Postsynaptic Cleft Presynaptic Axon Presynaptic: # of vesicles released, NT per vesicle, efficacy of reuptake mechanism. Postsynaptic: # of receptors, alignment & proximity of release site & receptors, efficacy of channels, geometry of dendrite/spine. Drugs: Prozac (serotonin reuptake), L-Dopa (NT in vesicles) Microtubule 7 / 29 8 / 29

3 Neurophysiology Balance of Electric and Diffusion Forces The neuron is a miniature electro-chemical system: 1 Balance of electric and diffusion forces. 2 Equilibrium potential. 3 Principal ions. 4 Integration equations. 5 The overall equilibrium potential. 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. We will use the equations describing this integration in our models. 9 / / 29 Electricity Resistance Ions encounter resistance when they move. Neurons have channels that limit flow of ions in/out of cell. + 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..) V + I The smaller the channel, the higher the resistance, the greater the potential needed to generate given amount of current (Ohm s law): G I = V R (1) Conductance G = 1/R, so I = VG 11 / / 29

4 Diffusion Equilibrium Constant motion causes mixing evens out distribution. Unlike electricity, diffusion acts on each ion separately can t compensate one + ion for another.. Balance between electricity and diffusion: E = Equilibrium potential = amount of electrical potential needed to counteract diffusion: I = G(V E) (3) (same deal with potentials, conductance, etc) (Fick s First law) I = DC (2) Also: Reversal potential (because current reverses on either side of E) Driving potential (flow of ions drives potential toward this value) 13 / / 29 The Neuron and its Ions Na+ Na/K Pump Excitatory Synaptic Input Cl +55 Inhibitory Synaptic Input Na+ 70 Cl K+ 70mV 0mV 70 Leak K+ Drugs and Ions Alcohol: closes Na General anesthesia: opens K Scorpion: opens Na and closes K Some kind of venom: closes all muscle firing (acetylcholine) Everything follows from the sodium pump, which creates the dynamic tension (compressing the spring, winding the clock) for subsequent neural action. 15 / / 29

5 V_m Putting it Together e = excitation (Na + ) i = inhibition (Cl ) l = leak (K + ). or I c = g c (t)ḡ c (V m (t) E c ) (4) 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 ) (5) V m (t + 1) = V m (t) dt vm I net (6) V m (t + 1) = V m (t) + dt vm I net (7) In Action I_net g_e = g_e =.2 30 V_m cycles (Two excitatory inputs at time 10, of conductances.4 and.2) 17 / / 29 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 (8) 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 e ḡ e g e ḡ e + g i ḡ i (9) Membrane potential computes a balance (weighted average) of excitatory and inhibitory inputs. Equilibrium Potential Illustrated Equilibrium V_m by g_e (g_l =.1) g_e (excitatory net input) 19 / / 29

6 Activation Computational Neurons (Units) Computational Neurons (Units) Overview V m = g g g g e e E + E + g g E e i i i l l l g g + g g e e i i + g g l l γ [ V m Θ] y + j γ [ V m Θ] net g e <x i w ij > + β = N 1 Really abstract: The standard sigmoidal function. 2 More neuro: The point neuron function. 3 Two kinds of outputs: discrete spiking, rate coded. x i w ij 1 Weights = synaptic efficacy; weighted input = x i w ij. 2 Net conductances (average across all inputs) excitatory (net = g e (t)), inhibitory g i (t). 3 Integrate conductances using V m update equation. 4 Compute output y j as spikes or rate code. 21 / / 29 Standard Sigmoidal Function Computing Excitatory Input Conductances First compute weighted, summed net input: η j = i x iw ij Then pass this through a sigmoidal function: y j = 1 1+e η j Sigmoidal Activation Function s a a+b Σ g e 1 < α x i w > ij x w i ij 1 N β A B Projections b 1 s a+b α x w i ij < x i w > ij Net Input One projection per group (layer) of sending units. Average weighted inputs x i w ij = 1 n Bias weight β: constant input. i x iw ij. Captures saturation (activation limits, nonlinearity) Misses V m dynamics (e.g., shunting inhibition). 23 / 29 Factor out expected activation level α. 24 / 29

7 activity Computing V m Thresholded Spike Outputs 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. 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 0 60 V_m In model: y j = 1 if V m > Θ, then reset (also keep track of rate). 25 / / 29 Rate Coded Output Convolution with Noise Output is average firing rate value. One unit = % spikes in population of neurons? Rate approximated by X-over-X-plus-1 ( x x+1 ): 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): y j = γ[v m(t) Θ] + γ[v m (t) Θ] (10) which is like a sigmoidal function: y j = (γ[v m (t) Θ] + ) 1 (11) compare to sigmoid: y j = 1 1+e η j γ is the gain: makes things sharper or duller. Θ 27 / / 29

8 Fit of Rate Code to Spikes spike rate noisy x/x V_m Q 29 / 29