Questions, comments and answers.
29 September 2010
Etymology You brought up a few etymological questions yesterday - and I have a pretty big interest in it so if you'd like to know:
Dendron is the Greek for tree.
Arbor is the Latin for tree.
And the stellate interneuron is named because of the Latin word 'stella' meaning star.
Stellar, right?
Reply Cool - thanks!
5 November 2010
Masters Is there any point at all in applying for a Masters if you don't have a II.1 or a 1st?
Reply
I amn't really the person to ask; I think it would be very hard to get into a research masters without a II.1 or 1st and the funding bodies don't allow it as far as I know. A taught master is different, then I don't think it would be so much of a problem.
9 April 2011
Voltage scale I was wondering if you could clarify something for me from past exam papers please. There is a question which asks you to "Derive the typical voltage scale of a neuron". This involves more (I assume) than saying that typically the voltage in the extracellular medium is defined to be 0mv and that inside is -70mv (which I know isn't deriving anything, just stating it). Is it the section in the book where the thermal energy necessary for an ion to overcome the negative potential of the membrane is described, using the Boltzmann probablility factor, exp { zqV/(k_b)(T) } ? And how this puts bounds on the extreme values the potential can take, i.e. from -90mv to 150mv? If so, is it necessary to describe the Nerst equation for this or is it sufficient to stop after describing the probability factor?
Reply
There is a different calculation this refers to, there is a heuristic calculation I did near the start of the course where I compared the thermal energy to the electromagnetic potential, before the chemical gradients were even introduced and noted that this indicated a voltage scale of 27 mV as a rough scale for the voltage dynamics of the neuron. It should be in your notes just before the stuff about the Nernst equation. Let me know if you can't find it.Further comment Thanks for getting back to me about this. I've found the piece that you mentioned, where eventually V_T = RT/F (in moles) = (k_B)T/q (for a single proton). That's the piece isn't it?
Reply That's the one.
14 April 2011
Sample Sorry to post this here, I couldn't find the comment link on the neuroscience page. Just noticed that in the previous years of the course there was a sample paper. Will you be posting one this year or should we just take the previous exams and sample papers as a guide to the style of question we should expect?
Reply
Aside from the stuff we didn't do because of the snow days, that is, models of learning in networks, the previous exams and samples are a very good guide to the exam
17 April 2011
Voltage scale Hi, i'm just a bit stuck on how to answer Q4b from the 2007 exam, "Derive an expression for the temporal receptive field of a neuron 'n terms of the signal response correlation and the signal auto-correlation.". i presume this is the work from 20th November, but i can't seem to find a single expression.
Reply
Sorry for the delay in replying, I somehow missed this message. This question is looking for a description of reverse correllation, the idea is that is you solve for the optimal kernel you get roughly Q_{rs}=D*Q_{ss} where the Q's are the response-stimulus and stimulus-stimulus correllations. To get D you ignore any objects, Fourier analysis both sides giving D(k)=Q_rs(k) / Q_rr(k) and then to an inverse transform. This is explained around p47 and p83 of the book.
18 April 2011
The synapse equation Regarding a question that has appeared on past papers and sample papers, say 2(b) on the 2009 paper. Ps -> Ps + Pm (1 - Ps). Show how this model can be justified. Someone asked about how to answer it before and you said: "You would be expected to give an brief explanation of this equation; basically if there is a fixed closing rate, each gate has a fixed chance of closing, then the total number closing is beta P and hence dP/dt=-beta P; or tau dP/dt=-P is tau=1/beta." Is that enough? What else could you say?Reply Sorry for the delay in replying, I somehow missed this message. Well the bit you quote explains the dynamics of P_s at general times, not at a spike, to solve for what happens when a spike arrives, in that case you need the opening equation, you assime alpha >> beta so dP/dt = alpha P during the opening phase, which lasts from 0 to T, you integrate to calculate how much the P_s changes by in this interval and then letting T-> 0 you get this equation.
24 April 2011
Nernst and Goldman equations I am emailing with a few questions on Neuroscience part 1. In the notes in class you derive the Nerst equation, do we need to know how to do this or do we just need to know the formula and result? Also, the book mentions the equilibrium potential being found by the Nerst equation whereas the reversal potential (I do understand the slight difference) is found by a slightly more difficult Goldman Equation. Is this important? Or do you just use the general term "reversal" potential to cover both? Is the Goldman equation therefore important for the purpose of our exam or not? It isn't mentioned in your class notes, at least, I haven't come across it yet. You have, however, summed up the different types of current, is this the same as the Goldman Equation? Sorry if my questions seem silly or trivial. Thank you for your help.Reply So it is important to know how to derive what I call the reversal potential, that is the equilibrium potential for each ion, which you calculate by applying the Nernst equation to each specific ion type; the difference between this and the books definition with its use of the Goldman equation wasn't covered.
25 April 2011
Sorry to bother you, but I had a quick query regarding linear models in the mathematical neuroscience course. I just wanted to ask if it is necessary to know the full derivation of the optimal linear kernel for estimating the firing rate for a given response; (i.e. the derivation that involves the use of functional differentiation or the calculus of variations), or is it merely necessary to know that the optimal kernel will minimise the error function between the actual and estimated firing rates and that this kernel is given by: Qrs(-tau)=D(tau)*Qss(tau) (where * denotes convolution)Reply Knowing the full derivation, however roughly would be good.
6 May 2011
3(c) on the 2007 exam asks you to "Show how this formula changes if there is a varying firing rate". Do you need to derive the formula again or just write down a new one/say what the difference is?
Reply A rough derivation showing where the differences come from would be needed for full marks
13 May 2011
2009 Q1d, Is the equation satisfied by n,m and h the Hodgkin Huxley equation with n^4 and m^3h inserted after the gk and gna respectively? Or is it three different relaxation equations with n,m and h? Thanks
Reply The wording seems to have caused some confusion, I meant three different relaxation equations with n,m and h; they satisfy the same type of equation
\dot{l}=\alpha_l (1-l) -\beta_l l
or equivalently
\tau_l \dot{l}= l_\infty
where l=n,m and h.
13 May 2011
circuit diagram Hi sorry another question. On question 1 (d) of the 2007 paper you asked to draw a circuit diagram for a cell- did we cover that this year?
Reply It was mentioned, but we didn't cover it like we did the year I asked that question.
13 May 2011
Linear? I just have a quick question about the equation governing a leaky integrate and fire neuron, is the model linear and how would you best justify this?
Reply The model isn't linear, if you have two solutions the sum isn't a solution because it wouldn't reset in the right place!
14 May 2011
Neuroscience Q2c 2007 Message=Hi, I was wondering if you could explain how to get the equation if there are 2 spikes? I can't find it in my notes. Thanks.
Reply That's given in the solutions, go to the problem sheet page, click on the link to 2008/9 and it's there, hopefully.
14 May 2011
STA "Show how the STA can be related to the kernel in a linear filter model of the firing rate" I've been tearing my notes apart trying to find something about this but don't seem to be able to find anything. I've found how the STA is related to the stimulus response correlation and for what circumstances the linear kernel is approximately equal to the STA, but nothing about the linear filter model. Could you possible shed some light?
Reply The linear filter model is \tilde{r}=r_0+K*s(t) where K is a kernel and s is the stimulus, we showed that this was proportional to the stimulus response correlation if the stimulus is white noise and hence it is proportional to the STA.
14 May 2011
Lots of questions
- When you ask why voltage dynamics are non-linear can we just say that the gates have non-linear dynamics and describe that a bit? What would be the ideal answer to this question? - Reply Assuming you mean the HH equation, yes then the ideal answer would be to say that the gates have strongly nonlinear dynamics, so, while the subthreshold behavior is reasonably linear, it becomes less linear when the dynamics of the voltage gated channels starts to have a significant effect and then, if there is spike, that is a highly nonlinear phenomenon.
- I'm just trying to make sure I have everything straight re different ions and their equilibrium potentials. There is a higher Na+ concentration outside the cell than inside, and higher K+ concentration inside than outside, right? The reversal potential for K+ is roughly -70mv and for Na+ it is c 50mv? Also, the potential is always the potential inside the cell where outside is said to be at 0, right - Reply Yes, that's all right.
- In both exams you asked to relate the concentrations inside and outside to the reversal potential. In this case, should we derive the Nernst equation or is it enough to write it down? - Reply Ideally you should give a rough derivation.
- If there is a single pre-synaptic spike what is the equation satisfied by the post-synaptic potential? Did we do this? I can't remember or find it in my notes. - Reply HH eqn with a synaptic current g(E_s-V) and then g is given by the equation in the start of the question.
- In that question (what equation does the PSP satisfy), does it depend on what P_s was before the spike? The equation for g will be g = g_max( 1 + [P_0 + P_max(1-P_0) - 1]*exp(-t/tau) ) ? Where P_0 is whatever P_s was before the spike arrived? - Reply Well if you are told there was only one spike then P_0=0, but the second spike has P_0=e^{-t_2/tau} or whatever.
- When you ask if spiking is a Poisson process, do you mean a homogenous Poisson process? - Reply No, spiking isn't even an inhomogeneous Poisson process, the refractory period rules that out.
- OK, why is that? Couldn't you have a rate function that took that into account? - Reply No, the rate function doesn't know when the spikes are.
- In one of the exams you asked how is the firing rate calculated and how is it calculated when there is a limited number of trials? I'm not sure what the difference is, isn't there always a limited number of trials? - Reply The fewer the trials the more useful it is to use some sort of clever kernel rather than histograms.
14 May 2011
Mnemonic I wouldn't dream of attaching my actual name to this, but I have come up with a silly mnemonic that might be useful for remembering that Sodium ions are more concentrated outside the neuron than inside and Potassium ions are more concentrated inside than outside (I guess this is under "resting conditions"? Anyway the mnemonic goes like this: We can assume that brains are clever (seems natural enough) so they would like to be healthy by not eating too much salt. Hence they keep sodium out. They also know that getting potassium in your diet is healthy (maybe by eating bananas or something) so they have potassium inside them. I guess that this is not perfect because there is such a thing as potassium poisoning. Though I hear you'd have to eat about 200 bananas to get a lethal dose of potassium.
Reply Useful mnemonics are always encouraged!
14 May 2011
Currents Is the leaky intergrate and fire model (Cm)dv/dt = -gl(V-El) + Ie/A and the hodgkin and Huxley is (Cm)dv/dt = -im (where im = terms with different ions)+ Ie/A? If not, could you please write them? I'm confused by all the different types
Reply Yes, plus the reset for the Integrate and Fire, often people divide across by g_l and write tau_m for C_m/g_l since it has the units of times.
15 May 2011
Nernst equation for Potassium I'm just a little confused about how to derive the Nernst equation for the Potassium ion. Since the concentration of potassium is higher on the inside, how does the Nernst equation work?
Reply You can still work out the reversal potential, the potassium ions want to get out so a negative internal voltage is needed to keep them in, in other words, have V negative inside the cell, work out the percentage of ions able to get out and balance it with the inflow. This is the example I do in the solution I put up last week, it is linked from the 2008/9 PS page which is in turn linked from this years PS page.
15 May 2011
In the notes,(Qss*D)(tao) = integral from 0 to infinity Qss(tao - tao')D(tao')dtao' and then it states that this is equal to sigma^2D(tao) and then D(tao)= Qrs(-tao)/ sigma ^2. I under stand we are using the white noise formula,and the fact that the convolution is equal to Qrs(-tao) but why is it no longer D (tao')?? If you didnt understand that last comment, its on page 34 of your scanned lecture notes . thanks
Reply So the white noise case is Q_{ss}(tao-tao')=sigma^2\delta(tao-tao') so the delta function gets rid of the integral and the tao'.
15 May 2011
2007 paperHi there, I was hoping you could help me with some questions I have about the 2007 paper. Firstly, Question 2(c),2007. I know that the equation for one spike will look like g=g_max e^{\frac{-t}{\tau_s}} but i'm not sure of what way to change the formula for a second spike arriving at t_2? For the eqn.
\tau_m \frac{dV}{dt}=E_I - V +g(E_a-V)
I assume all you need to do for this is sub in your value for each of the g's corresponding to the different spikes but again i'm not too sure. So i was wondering if you could tell me what happens here?
Also, and this may just be a simple matter of terminology, in Question 4,2007 you ask for an expression for the "temporal receptive field of a neuron" but we never use this expression in the notes. Are you referring to an expression for the Spike triggered average?
Oh and you also mention the "signal-response correlation" and the "signal auto correlation". I'm assuming that corresponds to the "response-stimulus correlation" and the "stimulus-stimulus correlation" that you mention in the notes.
I'm sorry if these seem silly and fairly obvious questions.
Reply So, yes, it is just a question of knowing the equation for g in both cases and knowing where it goes in the voltage equation, the one one you wrote down. So for one spike
g=g_max e^{\frac{-t}{\tau_s}}
is there is a second spike at t_2 then at that point P_s (t_2-), ie just before the spike, is
P_s(t_2-)=e^{-t_2/tau}
so using the equation in the earlier part of the question
P_s(t_2+)=e^{-t_2/tau}+(1-e^{-t_2/tau})P_{max}
and so
g=[g_max e^{-t_2/tau}+(1-e^{-t_2/tau})P_{max} ] exp(-(t-t_2)/tau)
which can be neatened at bit.
As for your other question, yes, that is a difference in terminology, I changed some of the terms I used after 2007. The receptive field is the kernel, so it isn't the same as the spike triggered average, but is related to it if the stimulus is white noise. Your questions aren't silly or obvious at all and even if they were, no need to apologize!
15 May 2011
ODEs/PDEs What command of ODEs and PDEs do you expect us to have? My ODEs are a little rusty because I haven't needed them for a year and a half and I never did PDEs.
Reply Well you need to be able to solve first order linear inhomogenous ODEs, you don't need PDEs for anything.
15 May 2011
Neuro: k=4 for Hodgkin-Huxley I can't see why this is so, maybe you can set me right on this? I can't see how to come to the conclusion that there are 4 independent, equally likely events necessary for a gate to be open. I can't see how the Hodgkin-Huxley equation is directly related to whether or not a gate is open.
Reply The gate has four independent subgates and all four subgates have to be open for the gate itself to be open, is say a gate has k subgates and each has a probability p of being open, then the chance all k are open is p^k if they are independent. The gates enter the HH equation through the conductivities in i_m: g_{Na} (E_{Na} - V) and g_{K} (E_{K}-V) and g_{K}=\tilde{g}_{K} n^4 for example, so n^4 gives the probability any gate is open and hence the proportion of gates that are open, \tilde{g}_{K} gives the total number of gates.
15 May 2011
Voltage Scale when you refer to the voltage scale of the neuron is it the equation 5.1 (page three of section 5.2) in Dayan and Abbot? ie V_T = ((R)(T))/F = ((k_B)(T))/q
Reply That's correct.
15 May 2011
STA, kernel etc More questions sorry! From question 4 on the 2007 paper. First of all what exactly is the linear convolution model? Is this just the equation for the predicted firing rate? And what would you state as the differences between the spike triggered average and the temporal receptive field (which is just the kernel yes?) Also how much are we expected to know about fourier transforms for the exam? I've never done anything about them before so are there some basics I should cover or is it enough to just know they can be used to calculate the kernel.
Reply The STA is approximately proportional to the stimulus response correlation, the optimal kernel, called the temporal receptive field in previous years, is only approximately proportional to the stimulus response correlation if the stimulus itself is white noise, otherwise in Fourier space you need to divide by the stimulus stimulus correlation. You need just enough Fourier analysis to do that calculation, alternatively that can be done by discretization and doing everything using matrices.
15 May 2011
Firing rate, inhomogeneous Poisson process Ok, last couple of questions I promise (maybe).
In the 2007 paper question 3(c)- there doesn't seem to be a formula for P[n] and a varying firing rate in my notes. Did we just not cover that this year or are you just looking for the probability of a given spike train P[t_1,....t_n] when there is a varying rate.
And question 3(d), how you calculate the firing rate. Is this the "moving window" formula? And what is done differently in the case of a limited number of trials? Cheers
Reply We did mention the variable rate version but perhaps without the emphasis and detail it received in other years, to calculate the P[n] we integrate P[t_1,t_2, . . . , t_n] over possible times. It comes out as the same as the constant version, but with rT replaced by \int_0^T r(t)dt
You should calculate the firing rate in principle by taking the delta t ->0 limit of \frac{1}{delta t} \int_t^{t+delta t}dt, but in practice there is only a finite amount of data and you use a moving window.
15 May 2011
Reversal Potential Vs Equilibrium Potential Every time I think I have this straight, I then see something exactly opposite to what I expect. Can you say what the difference between these is, please?
Reply So for the purpose of this course the reversal potential of a given ion is the potential when if the voltages were held fixed there would be no net flow, so it is an equilibrium potential for that specific ion. The equilibrium potential for the neuron is the voltage value where the voltage will be constant in the absense of any input. Effectively, since the gated channels have near zero conductivity near the equilibrium potential it is the reversal potential for the leak current.
16 May 2011
Poisson process, non-linearity hi, is neuronal spiking a poisson process? When drawing the electrical for a voltage compact neuron, can u explain why the corresponding voltage dynamics are not linear?
Reply No - the refractory period means the timing of a spike depends on previous spikes, that doesn't happen for a Poisson process. The voltage dynamics are not linear because the conductances depend on V so the Hodgkin Huxley equation has non-linear terms.
15 May 2011
Reversal Potential Vs Equilibrium Potential In my derivation for the formula for the Poisson process for spiking, I have two n! one above the line and one under in the fraction... one too many! where is this coming from? P_T [n] = lim (m-> infty) {(rt)^n / n! / n! } e^-rT Thanks!
Reply So if we have m slots thats P[n]= mCn (r T/m)^n (1-rT/m)^(m-n) where mCn is the binomial symbol m choose n, we choose n slots to put the m spikes into and we have the probability of choosing n slots and the probability of not choosing the other m-n. Now (1-rT/m)^m -> e^{-rT} and the (1-rT/m)^{-n} -> 1, finally the mCn = m!/(n! * (m-n)!) ->m^n / n! and this gives the result. Let me know if there is a bit that is confusing. There is actually an alternative derivation here, the one we did is shorter.
16 May 2011
Poisson process I understand how to derive the formula for a given spike train, but I can't figure out how to derive the formula for the probability of n spikes in a given period.
Reply You integrate out the times, the chance of n spikes is
(1/n!) int_0^T dt_1 int_0^T dt_2 . . . . int_0^T dt_n P[t_1,t_2, . . . ,t_n]
where you need to divide out by the 1/n! to deal with overcounting by reordering, ie counting (1,2) and (2,1) as two different spike trains.
23 May 2011
Part three I was desperately hoping to get a first this year and next year apply to do part three in cambridge, but I messed up my timing on a few exams and think it is more likely that I will come out with a two one, although it will hopefully be a decent one. If I still apply, will they laugh at me? Also, do you know of any decent universities that do a similar general-but-still-intense taught one year masters who perhaps wouldn't laugh at me? Thanks for having this, by the way, it's nice to have a place to ask questions which may be embarrassing in person.
Reply You should never worry about people laughing at you! Also, you should never try to do a selection committee's job for it, it is there descision whether to let you in, your to decide if you want to apply, so don't second guess the process, so you should certainly apply. As to whether you are likely to get a place with a II.1 and after a gap year, I don't know, I know when I did it they did let in a few people with II.1, but only a few and that was a million years ago. It also isn't clear yet that you did get a II.1, students are never very good at predicting how they did. As for other courses, I have lost touch, you need to ask someone who is still involved in your subject area, in TP Imperial used to have a good taught MSc, but I don't know if that's still the case and if there are other alternatives. Applying to the US is certainly another option.