Embed Size (px)
Citation preview
Models and Physiology of the Neuron
Literature
• C. Koch -‐ Biophysics of Computa=on • P. Dayan and L.F. AbboB -‐ Theore=cal Neuroscience • C. Koch and I. Segev (ed.) Methods in Neuronal Modeling
• G.M. Shepherd, The Synap=c Organiza=on of the Brain • S.H. Strogatz Nonlinear Dynamics and Chaos • E.M. Izhikevich Dynamical Systems in Neuroscience • J. Hertz, A Krogh, and R.G. Palmer, Introduc=on to the Theory of Neural Computa=on.
• W. Mass C. Bishop, Pulsed Neural Networks
Neuron
• Ramon & Cajal proposed the atomic theory of CNS in 1908
Neuron • More detail
Cell Membrane
Cell Membrane
Cell Membrane
Hodgkin and Huxley Model
Patch of cell membrane. In the model there is a capacitor C, to represent the membrane capacitance, a sodium conductance GNa, potassium conductance GK, and a leakage conductance GL. The membrane poten=al V is the poten9al inside the cell minus the poten=al outside
Hodgkin and Huxley Model
• Gna and Gk can be thought as the gate opening probability.
• NA channel has two types of gates (m and h) , and K just one.
Hodgkin and Huxley Model
• Changes in conductance and HH variables during an AP
Other Conductance Models
• These are the most accurate models for neural excitability
– FitzHugh-‐Nagumo – Morris-‐Lecar – Hindmarsh-‐Rose
The difficulty is that all these models are nonlinear and have mul=ple stable opera=ng points that depend on the model parameters
Dendri=c Tree Models
• Cable theory can be used to es=mate the conductance of the dendri=c tree as
where L is the electrotonic length of the cylinder which depends on its length, diameter, and resistance.
• A simple recursive algorithm scales linearly with the number of branches and can be used to calculate the effec=ve conductance of the tree.
• Compartment models are more accurate, but computa=onally more involved.
Synap=c models
The response of a neuron to individual neurotransmiBers can be modeled as an extension of the classical Hodgkin-‐Huxley model with both standard and nonstandard kine=c currents.
Four neurotransmiBers have primarily influence in the CNS.
AMPA/kainate receptors are fast excitatory mediators while NMDA receptors mediate considerably slower currents. Fast inhibitory currents go through GABAA receptors, while GABAB receptors mediate by secondary G-‐protein-‐ac=vated potassium channels.
14
A Wealth of Firing PaBerns
15
Neuron Models
There are many neuron models spanning a large range of complexity:
• Conductance based Models
• Hodgkin-‐Huxley • Compartment/Synap=c Models
• Threshold-‐Fire Model
• Simple Spiking Neuron Model (integrate and fire)
• Leaky Integrate-‐and-‐Fire Model
16
Spiking Neuron Model
Describe the state of ith neuron by a state variable ui. The neuron will fire at tk if u reaches a threshold θ. The set of all firing =mes is
Two different processes contribute to u:
• The state variable integrates the current over =me
• Aeer firing, the variable is reset by adding a nega=ve number
• The neuron may receive inputs from presynap=c neurons. The state is increased (or decreased) by
• To make it more accurate a refractory period can be add
17
Spiking Neuron Model
The problem is when the neuron receives excita=on below threshold that never decay!
Leaky Integrate and Fire
• In the leaky integrate-‐and-‐fire model a "leak" term is added to the membrane poten=al. The model looks like
where Rm is the membrane resistance. This forces the input current to exceed some threshold Ith = Vth / Rm in order to cause the cell to fire, else it will simply leak out any change in poten=al. The firing frequency thus looks like
which converges for large input currents to the previous leak-‐free model with refractory period
19
What is the Neural Code?
Two basic hypothesis:
• Pulse Timing Code – the arrival of each ac=on poten=al carries informa=on (evidence from the fly’s eye)
• Rate Code I – Number of spikes in a 100 –500 msec window (evidence from motor neurons).
• Rate Code II – Average over a neural popula=on. Very probably both methods co-‐exist.
20
Pulse Timing Codes
Define spike train as:
Where δ is the delta func=on and ti are the firing =mes.
A spike train contains only informa=on on the =mes the events occur. Informa=on w.r.t. s=mulus my be contained in the:
• Time to first spike
• Phase
• Correla=on and Synchrony
21
Rate Codes
Perhaps the most widely used quan=fier of spike trains is the “firing rate”, which measures the number of events per unit =me.
Several Measures:
The spike-‐count rate is the number (n) of spikes on a given interval T, or
From the spike train, let us count the number of spikes within a window, obtaining the mean firing rate
Where nsp is the number of spikes.
22
Rate Codes
Peri-‐S9mulus Time Histograms (PSTH)
Let us assume that one s=mulates repeatedly the same neuron, collect the data for each s=mulus, and then align the data. In each trial we can compute the spike-‐count average, and then average over the trials. This is called the trial average, denote here by < r >, and if we have sufficient trials one can decrease T1 to a rather small value, i.e. ΔT. Now we can define the firing rate r(t) as .
The Poisson Process • Define a =me interval [0,T], and assume we have n points to throw to
the =me interval. The ques=on is: what is the probability of gevng k of these n points land in a given sub interval t0 of [0,T]?
• Assuming that the experiment of throwing each point is independent of the others and each point we throw has a probability p of landing on the interval (p=t0/T), and probability q=1-‐ p of landing outside, the probability of k points falling in t0 is the binomial distribu=on
• Taking limits for n very large and the interval very small, we obtain
• where λ= n/T. If the interval t0 becomes infinitesimal (i.e. very very small) and we are only interested in the probability that a single point lands in the infinitesimal interval t0 (which becomes just a point t in the line) we can forget about the exponen=al and obtain . λ is called the rate (or intensity), and it measures the “density” of points in a macroscopic interval.
The Poisson Spike Train
• To go from points in the line to spike trains, we define a random process as an index set of random variables over =me. Specifically, let
If ti are random variables specified by the binomial probability law, this random process is called a Poisson process. What is interes=ng in the Poisson process is that both mean and the autocorrela=on func=on are completely specified by λ.
The Poisson Spike Train
• Note that now we have two different ways of specifying the spike train: by its =ming events or by its rate.
• They describe the process at two different scales – Timing methods – Rate methods
• Note that the rate assumes a model (memoryless in this case).
Signal Processing with Spike Train Single channel spike trains • Rate es=ma=on
τ controls the resolu=on and smoothness
Signal Processing with Spike Train Single channel spike trains • Spike trigger Average (what causes the spike?): Es=mates the mean value of the s=mulus over =me before a spike occurs. Several s=muli are presented and they are aligned with the occurrence of the spike. In a formula this reads
where the spike is assumed to occur at ti and x(t) is the input s=mulus
Signal Processing with Spike Train Single channel spike trains • Tuning curve (finer spike analysis): If the neuron is processing informa=on from the s=mulus, its firing rate will change when the s=mulus changes. Therefore count the number of spikes during the presenta=on of the s=mulus at each angle and then create a curve as a func=on of the parameter. The Gaussian tuning curve can be constructed as
where rmax is the largest firing rate observed, θmax is the maximal angle used (normally nega=ve and posi=ve) and σ is the width of the Gaussian (T is measured in Hz)
Signal Processing with Spike Train Single channel spike trains • Tuning curve for motor system (spike impact): The subject is engaged in reaching a target in a center out task (normally the hand) to different points arranged in a circle. Then the neuron firing rate is es=mated for each one of the direc=ons in the circle using the formula
The metric for evalua=ng the tuning property of a neuron is the tuning depth.
Signal Processing with Spike Train Pairwise Spike Train Analysis • Similarity between spike trains: Crosscorrela=on is mostly zero, so place a kernel over the spikes, effec=vely filtering the spike train with a linear filter. Oeen the impulse response of a first order linear system, where u(t) is the Heaviside func=on (zero for nega=ve =me, 1 for posi=ve =me) and τ controls the pole loca=on or the decay rate of the exponen=al. This yields so crosscorrela=on can be defined
Alterna=vely we can also measure the similarity over =me of a given spike train by compu=ng the autocorrela=on func=on
Signal Processing with Spike Train Pairwise Spike Train Analysis
Signal Processing with Spike Train Pairwise Spike Train Analysis • Dissimilarity measures: dis=nguish several different classes of neurons by their firing paBerns. Even though labels are not available clustering can s=ll explore the structure of the responses. The most widely used spike train distances are:
• van Rossum distance • The Cauchy Schwarz distance • Victor Purpura distance
Signal Processing with Spike Train Pairwise Spike Train Analysis • Van Rossum: This distance extends to spike trains the concept of Euclidean distance, so conceptually we are mapping a full spike train to a point defined as
Signal Processing with Spike Train Pairwise Spike Train Analysis • Cauchy-‐Schwarz distance: Another possible distance metric besides the Euclidean is the inner product distance between pairs of vectors (the cosine of the angle between the vectors). This concept can be generalized using the Cauchy Schwarz inequality, yielding
Signal Processing with Spike Train Pairwise Spike Train Analysis • Victor Purpura distance: evaluate the cost of transforming one spike train into the other. So it operates on the spike trains without the filter. Let us define the cost of moving a spike at tm to tn as q|tm − tn|, where q is a parameter expressing how costly the opera=on is. The cost of dele=ng or inser=ng a spike was set to one. Let us define
The VP distance between spike trains Si and Sj is defined as
where the minimiza=on is between the set of all unitary opera=ons c[l] that transform Si into Sj.
