Administrative Stuff

1. AdministrativeStuff • Labs: Metcalf 103;times: Wed and Thurs 5pm-7pm • Download sims(executables or source; see website) • All assignments (simulation exercises and RR) are due *before* class (1pm) on the date in syllabus. So you will only have this week’s lab to work onHW2. • Reading reactions: Better directly in email • CC Dan and Jasonon allreactions NO CLASS NEXT TUES 9/17

2. Neurons

3. Outline • Neuron overview: • Morphology, basic facts • Communication, synapses • Details: • Ion channels • Diffusion and electricity • Fick’s and Ohm’s laws • Ion flows at equilibrium, excitation, inhibition • Describing neuron and ion behavior mathematically • Neurons as detectors • What is a detector? • Content vs function • Components • Example uses • “Pandemonium” • Summary

4. Neuron overview

5. Overview: neuron morphology Purkinje cell (cerebellum) stellate cell (cortex) pyramidal cell (cortex)

6. Overview: neuron morphology RoLi Lab, Harvard

7. Neurons are detectors

8. You are a detector! fire! Thoughts Magic Senses suddenly orange colored... warmer than usual... smoking!

9. Neurons as detectors: actually Info to other neurons Spikes Morphology Synaptic input currents Info from other neurons

11. Neurons as detectors: actually (in dV/dt and also holistic senses)

12. Neurons as detectors: implications • Each neuron detects some set of conditions (e.g., smoke detector). • Neurons take each other’s outputs as inputs • For those interested: “Yesterday’s posterior is today’s prior!” • 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

13. Neurons as detectors: Examples What are some actual detector systems you are made out of? • sensory: detect bar of light, edges, tigers • motor: detect appropriate condition to move hand • abstract internal actions: engaging attention • regulation/homeostasis: detect too much overall activity Raises a certain question: What kinds of detectors “do” language, face recognition, creativity?

15. Neurons: how do they do it? For class: What’s a neuron? What do you know about them? • It’s a cell: body, membrane, nucleus, DNA, RNA, proteins, etc. • Ions (charged particles) present both inside and outside the neuron: Sodium (Na+), Chloride (Cl−), Potassium (K+) and Calcium (Ca++). Hence a brain is a mini-ocean. • Cell membrane has channels that allow ions (e.g. Na+) through. Channels can be open or closed (selective permeability). • When a neuron is at rest, greater concentration of negative ions inside the neuron vs. outside; the difference in charge is called the membrane potential (Vm). At rest it is typically around -70 mV.

16. Overview: neuron communication • Neurons communicate by firing “spikes” of electricity (action potentials) down their axons • When this current reaches the end of an axon, it triggers release of neurotransmitter into the synapse • Neurotransmitter binds to receptors in the receiving (postsynaptic) neuron, which opens dendritic synaptic input channels in the cell membrane potential of the postsynaptic neuron. • The flow of ions through these channels changes the potential of the postsynaptic membrane

17. Overview: synapses • Some synapses are excitatory • These mostly use glutamate • Glutamate binds to receptors and allows Na+ to enter the neuron, increasing Vm • Other synapses are inhibitory • These mostly use GABA • GABA binds to receptors and allows Cl− to enter the neuron, reducing Vm.

18. Overview: synaptic efficacy • Synaptic efficacy = how much is the activity of presynaptic (sending)neuron communicated to the postsynaptic (receiving) neuron • Presynaptic controls: # of vesicles released, NT per vesicle, efficacy of reuptake mechanism. • Postsynaptic controls: # of receptors, alignment & proximity of release site & receptors, efficacy of channels, geometry of dendrite/spine.

19. Details about ions etc.

20. Details: ions Some ions are more or less concentrated inside vs. outside the cell Positive and negative ions compete to set the overall ‘charge’

21. Net charge is a balance of diffusive and electric forces Diffusive force: • Shaking things around disperses them • Diffusion acts on each ion separately to make concentrations uniform Electrostatic force: • Ions are charged: Na+, Cl-, K+, Ca++. • Disparity in charges produces an electrical potential. • Current flows to even out distribution of + and - ions. • Ions encounter resistance when they move. • Neurons have channels that limit flow of ions in/out of cell. • The smaller the channel, the higher the resistance, the greater the potential needed to generate given amount of current (Ohm’s law).

22. Diffusive force: Fick’s (first) law Your kitchen smells like tacos and you open your bedroom door. How long does it take for your bedroom to smell like tacos? Depends on how strong the smell is and how quickly the molecules move. The strength is the concentration. The speed we call the diffusivity. We call the “amount of smell flowing into your room” the current. It turns out the currentis the concentration times the diffusivity. In symbols, I = -CD. Why does this make sense? By convention we add a minus sign, but don’t worry about it.

23. Electricity Ions have net charge: Sodium (Na+), Chloride (Cl−), Potassium(K+), and Calcium(Ca++). Positive and negative charge (opposites attract, likerepels). Current flows to even out distribution of + and -ions. Disparity in charges produces potential (the potential to generatecurrent).

24. Electric Force: Ohm’s Law Ions encounter resistance when theymove. (Analogy: water flowing down a rocky stream) Neurons have channels that limit flow of ions in/out ofcell. The smaller the channel, the higher the resistance, the greater the potential needed to generate given amount of current (Ohm’slaw): V R I= Conductance G = 1/R, so I =GV

25. Equilibrium: Balance between electricity anddiffusion E = Equilibrium potential = amount of electrical potential neededto counteract diffusion: I =G(V − E) E is the electric potential at which the diffusion force would pull current in the opposite direction with equal force. I flows in proportion to voltage difference fromequilibrium.

26. Equilibrium: Balance between electricity anddiffusion E = Equilibrium potential = amount of electrical potential neededto counteract diffusion: I =G(V − E) E is the electric potential at which the diffusion force would pull current in the opposite direction with equal force. I flows in proportion to voltage difference fromequilibrium. Other terms forE: Reversal potential (because current reverses on either side ofE) Driving potential (flow of ions drives potential toward thisvalue)

27. Each ion has it’s ownequilibrium “Eq potential for Na ENa: If sodium had its way, the neuron would settleto into this steady state without any otherforces” For each ion, E is proportional to concentration outside/inside ofcell: E>0whenconcentrationhigheroutside,andE<0when higher inside (Nernst equation). RT[Xo] nF [Xi] log (7) E=

28. Na/K pump: winding the spring • Redistribution of ions by diffusion and electrical potentials would make the brain homogeneous (stuck in one state) • Maintaining different conditions inside and outside neurons requires energy: “active” transport. • Sodium-potassium pumps are molecules in the cell membrane which push Na + out of the neuron in exchange for a lesser amount of K + entering. The result is a net loss in charge. • This creates a dynamic tension in the cell: When the neuron is at rest, Na + wants to come back in (because of both electrical and diffusion forces), but it can’t because the Na channels are closed!

29. Details: equilibrium • Cell is around -70mV • Na+ wants in • Diffusion pushes in • Electrical pushes in • Cl− is in balance • electrical pushes out • diffusion pushes in • K+ is in balance • electrical pushes in • diffusion pushes out

30. Details: excitatory input • Na+ rushes in, making potential more positive • If input continues, Na+ inflow continues until membrane potential reaches +55mV • This is the reversal potential for Na+ • Cl− wants in, but can’t (channels closed) • K+ starts to leak out

31. Details: inhibitory input If Vm = - 70 mV • nothing happens If Vm > -70mV • Cl- starts to come in • This counteracts any influx Na+

32. It’s Just a LeakyBucket excitation ge Vm inhibition/ leak gi/l ge = rate of flow intobucket gi/l = rate of “leak” out ofbucket Vm = balance between theseforces

33. Or aTug-of-War excitation ge Ee Vm Vm Vm Vm Ei gi inhibition

34. Summary: excitation, inhibition, leak • Excitatory synaptic input boosts the membrane potential by allowingNa+ ions to enter the neuron • Inhibitory synaptic input serves to counteract this increase inmembrane potential by allowing Cl− ions to enter the neuron • The leak current (K+ flowing across the membrane through open channels) acts as a drag on the membrane potential. Functionallyspeaking, it makes it harder for excitatory input to increase themembrane potential.

35. Mathematical descriptions

36. Math: Why do it? Refresher on models: Why? • Qualitative hypotheses are only sort of hypotheses. • Mathematical models are completely explicit. • Your intuition is limited. Simulation of neurons: • Neural activity (and learning) can be characterized by equations. • These specify the behavior of artificial neurons. • Artificial neurons can then be put together to explore behaviors of networks which you probably wouldn’t have realized were implied.

37. Math: equilibrium potential • Want to describe total current, not just components • At equilibrium, concentrations don’t change, implying forces cancel • The equilibrium potential E is that which exactly counters diffusion • Hence, if actual voltage higher or lower, there will be a current. current = conductance*( equilibrium potential - actual voltage ) • I.e: Current flows in proportion to voltage difference from equilibrium. • Other terms for E: • Reversal potential (current reverses passing E) • Driving potential (flow of ions drives potential toward this value)

38. Math: currents • Each ion obeys a current equation like this: I = g(Erev. - Vm) • One such current is associated with Na+, one with Cl-, one with K+. Using e, i, and l for excitation, inhibition, and leak, we can write: Itotal = Ie + Ii + Il Itotal = ge(Ee- Vm) + gi(Ei- Vm) + gl(El - Vm)

39. Putting itTogether Ic=gc(Ec−Vm) e = excitation(Na+) i = inhibition(Cl−) l = leak(K+).

40. Putting itTogether Ic=gc(Ec−Vm) e = excitation(Na+) i = inhibition(Cl−) l = leak(K+). = ge(Ee −Vm) + gi(Ei−Vm)+ gl(El −Vm) Inet Vm(t+1)=Vm(t)+dtvmInet

41. Putting it Together: WithTime Ic=gc(t)g¯c(Ec−Vm(t)) e = excitation(Na+) i = inhibition(Cl−) l = leak(K+). = ge(t)g¯e(Ee −Vm(t))+ gi(t)g¯i(Ei−Vm(t))+ gl(t)g¯l(El −Vm(t)) Inet Vm(t+1)=Vm(t)+dtvmInet g(t) : time-varying; number of open channels for each ion at any time g_bar: maximal strength (conductance) of these channels dt_vm: time constant of how quickly Vm can change (determined by capacitance of cell membrane)

44. Differential equation version (common in computationneurosci) dVm Cm = ge(t)g¯e(Ee −Vm) + gi(t)g¯i(Ei−Vm)+ gl(t)g¯l(El −Vm) + ... dt • Cm = membrane capacitance, determined by surface area ofmembrane • holds charge; reduces speed at which voltage can change(dtvm)

45. Differential equation version (common in computationneurosci) dVm Cm = ge(t)g¯e(Ee −Vm) + gi(t)g¯i(Ei−Vm)+ gl(t)g¯l(El −Vm) + ... dt • Cm = membrane capacitance, determined by surface area ofmembrane • holds charge; reduces speed at which voltage can change(dtvm) Equivalentcircuit. timeconstantτ=RC=C/gnet

46. How Does Neuron “Decide” When toSpike? • When membrane potential exceeds a threshold value, voltage-gated Na+ channels openup • This leads to an influx of Na+ and (consequently) a very large and rapid increase in membrane potential • Shortly afterward, voltage gated K+ channels openup • This leads to a rapid flow of K+out of the neuron and thus a very large and rapid decrease in membranepotential • The result is a discrete “spike” in membrane potential

47. Spike = ActionPotential This travels down the axon in a waveof activity...

50. Rate Code approximates rate produced by spiking model