Spatial extent of neurons: dendrites and axons 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 white matter, 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 1 @h τ (V ) = −h + h (V ) h @t 1 @n τ (V ) = −n + n (V ) n @t 1 • Particular example of a reaction-diffusion system • Travelling wave solution(s) V (x − ct; t) = V^ (x) i.e. a spike travelling along the axon with constant speed c p • Speed c / d - thicker axons propagate spikes faster. Travelling wave solution to HH equations Myelinated axons • In myelinated region, no channels are present; • Membrane potential 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 myelin 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 action potential 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 dendrite 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 cable theory • 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.