Lecture 11 : Simple Neuron Models Dr Eileen Nugent
Reading List Nelson, Biological Physics, Chapter 12 Phillips, PBoC, Chapter 17 Gerstner, Neuronal Dynamics: from single neurons to networks and models of cognition (CUP 2014), Chapter 1 & Chapter 5 A.L. Hodgkin & A.F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerves. J. Physiol. (Lond.) 117, 500-544. (1952) Edward S Boyden et al, Millisecond-timescale, genetically targeted optical control of neural activity, Nature Neuroscience (2005) L Badel et al, Dynamic I-V Curves Are Reliable Predictors of Naturalistic Pyramidal-Neuron Voltage Traces, Journal of Neurophysiology, 99, 656-666 (2007) 2
Action Potential Measurements ~ 500 um 2 ms Smithsonian National Museum of Natural History - Research C Roper 6 feet next to giant squid Hodgkin Huxley 1939 Giant axon of squid Features of action potential : Change sign, overshoot
Hodgkin-Huxley Experimental Apparatus Experimental Aims : Determine the voltage dependence of axon membrane conductance Space Clamping Squid axons were large enough for a long wire to be threaded through them which allowed their interior to be maintained at a spatially uniform electric potential throughout length of axon. This eliminates voltage gradients and hence propagation. Voltage Clamping A Feedback circuit monitors V the transmembrane voltage and adjusts current I to maintain V at a command Voltage set by the experimenter Separation of the effects of different ion currents is possible by eliminating ionic flows from ion species other than the species of interest. Elimination is achieved by adjusting the external concentration of an ion species such that the Nernst potential for that species is equal to the command voltage. 4
Hodgkin - Huxley Measurements Total Membrane Current The sodium current contribution is a transient inward current. The peak of the current tracks the command potential Potassium conductance rises to a voltage dependent value and remains at this value when V is held fixed
Hodgkin Huxley Proposed a basic model of nerve membrane as consisting of independent channels containing gates. Currents carried by ions moving down electrochemical gradients. Obtained the parameters for the model : characteristics of channel types, gates and voltage dependency of gating. Numerically integrated the equations and comparison of predicted membrane potential with the shape of action potential in space clamped neuron 6
Hodgkin-Huxley Equations For a reasonable description of an action potential need to add to the cable equation the inactivation of sodium channels - intermediate non-conducting state from which the channel can t be opened Hodgkin-Huxley Model - more detailed including voltage dependence of potassium conductance - mathematical formalism C dv dt = g m (V E m ) g Na m 3 h(v E Na ) g K n 4 (V E K ), t x (V ) dx dt = x + x (V ), gating variables m,n and h - model the probability that a channel is open at a given moment in time and tend to a stationary value (x ) with time constant (τ) m,h activation/inactivation Na channels ; n activation of the K channels ENa, EK - Nernst potentials; Em resting membrane potential
Action Potential Generation Molecular mechanism for voltage gated ion conductances is ion channels which are proteins embedded in the membrane which open/close in a voltage dependent manner Cell membranes ability to conduct sodium is voltage dependent Membrane patch: bistable switch between V K nernst and V Na nernst depending on whether sodium channels are open or not Depolarisation of one patch raises the neighbouring patch above threshold giving a spike of membrane depolarisation down the axon at finite speed
Voltage gated sodium channel External field ε makes contribution qεd to free energy b membrane thickness V/b electric field in membrane d distance moved by charged component of channel
Patch-clamp apparatus used to record current through single (few) channel Channels artificially kept open found to be highly selective Results for sodium channel Patch Clamping
Patch clamp measurements on Sodium Channels (a) Depolarizing voltage step applied (b) Nine individual current responses (potassium channels blocked, patch contained 2-3 active channels) (c) Average over 300 individual recordings
Voltage Gating Expt under Voltage clamp conditions Fit to data Can be modeled as a two-state system Snaps between two different conformations
Information Encoding The action potential (spike) is the elementary unit of signal transmission Form of the spike does not carry information - essentially indistinguishable Number and the timing of spikes which matter. cell 1/cell 2 from different animals respond with remarkable similar spike trains to a visual stimulus - is there a model that can predict this response?
bursting adapting low Multiple Firing Patterns INITIATION PATTERN tonic low high STEADY-STATE PATTERN low high high 14
Integrate and Fire Models Studies of of neural coding, computation & learning rely building block: the single neuron Mathematical models of different complexity built to understand the response of the neuron to a stimulus Complex biophysical models like HH - Model many spatial and temporal details of ionic flow - Easy to relate model to Biological processes in the neuron - Too computationally expensive to be used in large networks for modelling Search for simpler models which are accurate enough to capture the key dynamic properties of the cell - Integrate and fire models - Harder to relate to Biological processes in the neuron - Tradeoff between speed and accuracy
Perfect Integrate and Fire Neuron accumulates input current I(t) by increasing membrane voltage V(t) Spike emitted when it hits a threshold and the membrane potential is reset to Vr dv = 1 dt C I(t) when V (t) > V T then V (t)! V r. r V (t)=v r + 1 C Z t t 0 I(s)ds.
κ (GΩ/μs ) Leaky Integrate-and-Fire j i Spike emission V Vr I C g L E 0 Vr reset C dv = g L (V E 0 )+I(t) Current injected (4) can either dt accumulate on the membrane or when V (t) > V T then V (t)! V r. leak out (5) the Kirchoff law for conservation of charge. The current injected
Leaky Integrate-and-Fire Response to Constant Input Current Repetitive, current I0 V T t 1/T frequency-current relation V Repetitive, current I1> I0 f-i curve I t
Non-linear Integrate and Fire Models C dv = F(V )+I(t) dt where F V F(V )=0 can. Forspecul the perfect IF it is (F(V )= g L (V E 0 ) mplest of the non-linearity. The w simple F(V )= g L (V E 0 )(V V T ) he dynamics at hyperpolarized p V F(V )= g L (V E 0 )+g L D T exp D T VT Allowing the transmembrane current to be any fn of V : Perfect LIF - linear negative slope Quadratic Exponential 19
Refractoriness Refractoriness: second spike prevented immediately after one was emitted. Absolute and relative refractory periods (~ 1ms). Duration of the spike is often taken as the absolute refractory period since it is impossible to emit a spike while one is being triggered. Relative refractory period it is possible to fire a spike, but a stronger stimulus is required. In this case the current required depends on the time since the last spike. Absolute refractory period always precedes the relative refractory period, and the absolute refractory period can be seen as a very strong relative refractory period. Due to inactivation of sodium channels
Measurements of the Non-Linearity E L V T I(t) Fig. 8 L Badel et al, Dynamic I-V Curves Are Reliable Predictors of Naturalistic Pyramidal-Neuron Voltage Traces, Journal of Neurophysiology, 99, 656-666 (2007) 21
Theory versus experiment for exponential integrate and fire model with experimentally tuned parameters V NLIF with F(V) measured No Adaptation, Noise, Effects of dendrites or synapses L Badel et al, Dynamic I-V Curves Are Reliable Predictors of Naturalistic Pyramidal-Neuron Voltage Traces, Journal of Neurophysiology, 99, 656-666 (2007) 22
Optogenetics TECHNICAL REPORT ishing Group http://www.nature.com/natureneuroscience Millisecond-timescale, genetically targeted optical control of neural activity Edward S Boyden 1, Feng Zhang 1, Ernst Bamberg 2,3, Georg Nagel 2,5 & Karl Deisseroth 1,4 Temporally precise, noninvasive control of activity in welldefined neuronal populations is a long-sought goal of systems neuroscience. We adapted for this purpose the naturally occurring algal protein Channelrhodopsin-2, a rapidly gated light-sensitive cation channel, by using lentiviral gene delivery in combination with high-speed optical switching to photostimulate mammalian neurons. We demonstrate reliable, millisecondtimescale control of neuronal spiking, as well as control of excitatory and inhibitory synaptic transmission. This technology allows the use of light to alter neural processing at the level of single spikes and synaptic events, yielding a widely applicable tool for neuroscientists and biomedical engineers. in intact brain tissue have provided many insights into the function of circuit subfields (for example, see refs. 1 3), neurons belonging to a specific class are often sparsely embedded within tissue, posing fundamental challenges for resolving the role of particular neuron types in information processing. A high temporal resolution, noninvasive, genetically based method to control neural activity would enable elucidation of the temporal activity patterns in specific neurons that drive circuit dynamics, plasticity and behavior. Despite substantial progress made in the analysis of neural network geometry by means of non cell-type-specific techniques like glutamate uncaging (for example, see refs. 4 7), no tool has yet been invented with the requisite spatiotemporal resolution to probe neural coding at the resolution of single spikes. Furthermore, previous genetically 23
Light-Gated Ion Channels 24
Light Driven Neural Spiking and Inhibition 25
Pulse Trains 26
Control of Brain Circuits 27