Spatial extent of : and Morphological diversity of neurons Cortical pyramidal cell Purkinje cell Retinal ganglion cells Motoneurons Dendrites: spiny vs non-spiny Recording and simulating dendrites Axons - myelinated vs unmyelinated Axons - myelinated vs unmyelinated

• Myelinated axons:

– Long-range axonal projections (motoneurons, long-range cortico-cortical connections in , etc)

– Saltatory conduction;

– Fast propagation (10s of m/s)

• Unmyelinated axons:

– Most local axonal projections

– Continuous conduction

– Slower propagation (a few m/s) Recording from axons Recording from axons Recording from axons - where is the spike generated? Modeling neuronal processes as electrical cables

• Axial current flowing along a neuronal cable due to voltage gradient:

V (x + ∆x, t) − V (x, t) = −Ilong(x, t)RL ∆x = −I (x, t) r long πa2 L where

– RL: total resistance of a cable of length ∆x and radius a;

– rL: specific intracellular resistivity

• ∆x → 0: πa2 ∂V Ilong(x, t) = − (x, t) rL ∂x The cable equation

• Current balance in a cylinder of width ∆x and radius a

• Axial currents leaving/flowing into the cylinder πa2 ∂V ∂V  Ilong(x + ∆x, t) − Ilong(x, t) = − (x + ∆x, t) − (x, t) rL ∂x ∂x • Ionic current(s) flowing into/out of the cell

2πa∆xIion(x, t)

• Capacitive current ∂V I (x, t) = 2πa∆xc cap M ∂t • Kirschoff law

Ilong(x + ∆x, t) − Ilong(x, t) + 2πa∆xIion(x, t) + Icap(x, t) = 0

• In the ∆x → 0 limit: ∂V a ∂2V cM = 2 − Iion ∂t 2rL ∂x Compartmental appoach Compartmental approach Modeling passive dendrites: Cable equation for a passive membrane

• For a passive membrane (taking the resting potential to be zero) ∂V a ∂2V V cM = 2 − ∂t 2rL ∂x rM or ∂V ∂2V τ = λ2 − V M ∂t ∂x2 where

– τM = cM rM is the membrane time constant; q – λ = arM is the space or length constant 2rL – Electrotonic length of a cable: l/λ where l is length of the cable Steady-state solution

• Steady-state voltage ∂2V λ2 − V = 0 ∂x2 • Example: semi-infinite cable x ≥ 0,

current I0 applied at x = 0 ⇒ Boundary condition: πa2 ∂V I0 = − (x = 0) rL ∂x • Full: sealed end Solution • Dashed: open end λrL  x V (x) = I0 exp − πa2 λ • Dotted: semi-infinite Attenuation is frequency dependent Modeling axons: Spatially extended Hodgkin-Huxley model

• Spatially extended Hodgkin-Huxley model

2 ∂V d ∂ V 3 CM = 2 − gL(V − VL) − gNam h(V − VNa) − gK (V − VK ) + I ∂t 4Ri ∂x ∂m τ (V ) = −m + m (V ) m ∂t ∞ ∂h τ (V ) = −h + h (V ) h ∂t ∞ ∂n τ (V ) = −n + n (V ) n ∂t ∞

• Particular example of a reaction-diffusion system

• Travelling wave solution(s) V (x − ct, t) = Vˆ (x)

i.e. a spike travelling along the with constant speed c √ • Speed c ∝ d - thicker axons propagate spikes faster. Travelling wave solution to HH equations Myelinated axons

• In myelinated region, no channels are present;

• satisfies the diffusion equation

2 2 cmy ∂V πa1 ∂ V = 2 L ∂t RL ∂x Myelinated axons

• Membrane potential satisfies the diffusion equation

2 2 cmy ∂V πa1 ∂ V = 2 L ∂t RL ∂x

where the total capacitance cmy is much smaller than an unmyelinated membrane,

cmy ∼ CM 2πdmL/log(a2/a1) where dm is the thickness of one of the cell layers composing the sheet.

• Optimizing diffusion coefficient leads to a1 ∼ 0.6a2, similar to experimentally observed values;

•⇒ Velocity of propagation scales linearly with diameter, unlike in unmyelinated axons Myelinated axons

• At nodes of Ranvier, high density of sodium channels, so is ‘regenerated’

• Spike jumps from node to node, with a higher velocity than in unmyelinated axons (by approximately a factor 10) Active dendrites

Active voltage-gated conductances lead to non-linear integration of inputs

• NMDA channels

• Calcium channels

• Sodium channels BAC firing

• Backpropagation-activated Ca2+ (BAC) spike firing, depends on coincident synaptic input to the tuft with a single BPAP at the cell body

• Triggers burst of action potentials Bursting due to active dendrites: Pinsky-Rinzel (1994) Multiplicative interactions between inputs due to active dendrites

• Gain of frequency vs somatic input curve increase with dendritic inputs

• Multiplicative interactions between dis- tal and proximal inputs Multiple layer models of single neurons Dendrites

• Can be described using

• Classic view: dendrites are passive devices, lead to linear summation of synaptic inputs

• Emerging view: dendrites are potentially active devices, leading to non-linear summation of inputs Bibliography

• Dayan and Abbott, chapter 6;

• Ermentrout-Terman book, chapters 2 and 6

• Koch book, chapters 2,3

• Johnston and Wu book, chapter 4

Recent review papers

• Branco and Hausser, Current Opinion in Neurobiology 2010

• Silver, Nature Reviews Neuroscience 2010

• Major, Larkum and Schiller, Annual Reviews Neuroscience 2013

• Larkum, Trends in Neurosciences 2013