Exercise 1 : Digital vs. Analog

Transcription

1 Biophysics of Neural Computation, Introduction to Neuroinformatics (Winter 2004/2005) Exercise 1 (1) Here we show you a typical Purkinje cell of the cerebellum, you have already encountered in the lecture. Biophysics of Neural Computation You easily identify: the Introduction soma, the dendrites to and Neuroinformatics the axon: WS Prof. Rodney Douglas, Kevan Martin, Hans Scherberger, Matthew Cook Ass. Frederic Zubler fred@ini.phys.ethz.ch fred/bnc.html Welcome to the exercises of "Introduction to Neuroinformatics"! Since this course covers different topics and you all have a different background, some parts will seem easy, whereas some others will be much more difficult, depending on your training (obviously a biologist, a physicist or a computer scientist won t agree on which ones are the difficult ones...). Exercise 1 : Digital vs. Analog Biophysics of Neural Computation, Introduction to Neuroinformatics (Winter 2004/2005) Exercise 1 1. In Fig 1 you see two different neurons. Identify the soma, the dendrites and the axon. Which one (1) Here of we the show processes you a typical(axon Purkinje Whichor cell one dendrite) of of the the cerebellum, processes corresponds you (axon have already dendrite) toencountered the corresponds input the toand lecture. the input which and one whichto onethe to the output of You easily of the identify neuron? the soma, the What dendrites the neuron? areand the thefunctional axon: consequences of these differences in morphology? Below we show you a different type of neuron (a bipolar cell from the retina). (Think of the different computations these neurons have to perform). Which one of the processes (axon or dendrite) corresponds to the input and which one to the output of the neuron? Figure 1: (left) a typical Obviously Purkinje the morphology cell of the of this cerebellum neuron differs (right) a lot from a bipolar the Purkinje cell from cell. What the retina. are the functional Below we show you a different typeconsequences of neuron (a bipolar of these celldifferences from the retina). morphology? (Think of the different computations these neurons have to perform) 2. In Fig 2 we look at the signals observed at the different parts of a sensory neuron responding to stretch, involved in the knee jerk reflex.. (a) On the leftmost panels different stimuli to our neuron are depicted. On the y-axis the stimulus strength is plotted, on the x-axis its duration. As mentioned, for this particular neuron the stimulus is a stretch of the muscle it is attached to. What could be the use to have stretch receptors in our muscles? What other types of stimuli can you think of (for different sensory neurons)? (b) At the receptive surface of a sensory neuron certain proteins are selective to the sensory stimulus. Those form ion channels regulating the ion-flow through the membrane. This results in a change of the membrane potential ("receptor potential") as depicted in the second diagram from the left in each line. What happens to the receptor potential if you double the amplitude of the stimulus? If you double its duration? Is the receptor potential an analog (graded) or digital (all-or-none) signal? Obviously the morphology of this neuron differs a lot from the Purkinje cell. What are the functional consequences of these differences in morphology? (Think of the different computations these neurons have to(c) perform) Now look at the third figure from the right. If the membrane potential exceeds a certain voltage threshold at the trigger zone, action potentials (spikes) are generated, which propagate along the axon. (We will learn in detail how this spike generation works in terms of membrane properties in later lectures). How is the signal transformed? What would happen if you reduce the amplitude of the stimulus to a tenth of the original strength? Sketch the firing frequency (the number of spikes in a given interval of time) as a function of stimulus amplitude. Does the line you drew intersect the origin?

2 Why or why not? (d)on the axon the only signal you see is the train of action potentials. Can you explain why this signal is both analog and digital? (e) When the signal arrives at the synaptic bouton the electrical signal is transferred into a chemical one through release of neurotransmitter molecules. How does the amount of transmitter released depend on the number of spikes? How does it depend on the initial sensory stimulus? Is this signal analog or digital? (f) Now look at the whole system from stimulus to synaptic bouton. What kind of computation/operation does it perform? Why does it make sense to use analog and digitial signals in (2) the Indifferent this exercise stages? we look at the signals observed at the different parts of a sensory neuron responding to stretch, involved in the knee jerk reflex. Figure 2: A typical peripheral sensory neuron. (a) On the leftmost panels different stimuli to our neuron are depicted. On the y-axis the stimulus strength is plotted, on the x-axis its duration. As mentioned, for this particular neuron the stimulus is a stretch of the muscle it is attached to. What could be the use to have stretch receptors in our muscles? What other types of stimuli can you think of (for different sensory neurons)? Exercise 2 : Signal transmission in a network We will consider a neuron as a unit that can be be active or inactive. Each unit is linked to some others by synapses, and can influence them to become -or not- active. In this exercise we are interested in how a signal (the activation of neurons) can be transmitted, amplified or refined as it spreads across a network. A) The avalanche model : The avalanche (auf deutsch: Lawine) model is defined by the following rule: whenever a neuron is activated, it (b) At the receptive surface of a sensory neuron certain proteins are selective to the sensory stimulus. Those activates form each ion of channels its n postsynaptic regulating the neurons ion-flow with through a probability the membrane. p. So in This average results an in active a change neuron of the activates membrane np neurons potential on the ( receptor next step. potential ) The same as rule depicted is then in applied the second to any diagram of the from neurons the left that in each were line. activated, and so on. (Fig 3 What - left(a) happens ) to the receptor potential if you double the amplitude of the stimulus? If you double its duration? Is the receptor potential a graded (analog) or all-or-none (digital) signal?

3 a) At time 0 we start with one neuron active. At time 1 we have np neurons active. How many neurons are active at time 2? and at time t? Compare your result with Fig 3 - left(b). b) Can you explain Fig 3 - left(c) and Fig 3 - left(d)? c) If you think of the sensory neuron described in Exercise 1, why does its signal need to be amplified at some point? d) In the figure we have only feed-forward connections. What is the implication if neurons can activate other neurons in the same layer, or even in the previous layers? B) The synfire chain Now we consider groups of neurons coupled with an all-to-all connectivity in the feed-forward direction shown in Fig 3 - right(a). Let us use a more realistic model where the time is continuous (and the transmission of the signal from one to the next cell takes roughly 5ms). One cell alone is not enough to activate a neuron of the next layer (i.e.) several simultaneous neurons connecting the same post-synaptic cell muss be active to excite the next one. Morevover, some random noise is added. In these conditions, what additional mechanisms than the critical level of initially active cells influence the success of the signal propagation? Figure 3: (Left) avalanches (Right) synfire chains Exercise 3 : Generation of a complex pattern In biological neural systems, a cluster of dense neurons is called a nucleus. In Fig 4 you can see the brain of a bird, the zebra finch, in which three nuclei, named RA, HVC and X are shown. These nuclei are involved in the production of the bird song, but they don t directly contact the muscles responsible for the production of the song, and therefore are called pre-motor. Under the brain you see the brainstem, and the nucleus (nxii) from which originates the nerve that controls the muscles producing the song. a) How can we demonstrate that a specific brain region is involved in a particular task? In HVC there are at least three different types of neurons. The first are projecting onto the neurons in the nucleus X. The second type project onto RA. The third stay in HVC, and are called interneurons. The zebra finch is famous for having a specific song motif that it keeps repeating (always the same). Fig 5 shows a spectrogram of the song motif for a particular individual. (Next to it you see the spectrogram for a

4 call to a female). Below are shown activity of neurons recorded in HVC. Each row of tick marks shows spikes generated during one rendition of the song or call; roughly ten renditions are shown for each neuron. Neural activity is aligned by the acoustic onset of the vocal motif. b) What differences do you observe between the neurons projecting onto RA (HVC RA )and the neurons that only contact other cells in HVC (interneurons)? c) What do you think is the effect of the loss of one neuron in HVC or RA? And in nxii? and the loss of one vocal muscle? d) As we have said, the song is stereotyped in the zebra finch (always the same pattern of vocalization is produced). Can you think of any stereotyped motor activity in humans? Figure 4: A bird (Zebra finch) brain Figure 5: Spectrogram and raster plots Exercise 4 : Diffusion We start with a membrane separating two solutions (ions, molecules and water), electrically neurtal. Suppose that the membrane is only permeable to one type of ion. If we add a certain quantity of this ion (of course with another non-diffusible ion of the opposite charge) on one side of the membrane, so that the concentration

5 becomes higher in the compartment A than in B, there will be a net diffusion of this ion [from A to B][from B to A]. If our ion carries a positive charge, the difference of concentration will result in an electrical potential difference across that membrane, with side A being more [positif][negatif]. 2 forces of [opposite][the same] direction act on the ion: the chemical and the electrical one. Exercise 5 : The Nernst Equation The Nernst equation gives the voltage E ion that balances a given difference in chemical concentration across a cell membrane ([Ion] intracellular and [Ion] extracellular ): E ion = RT zf ln([ion] extracellular [Ion] intracellular ) where (E ion is measured in volts, R is the gas constant [8.31 joules / degree / mole], T is the absolute temperature, F is the faraday constant [96, 500 coulombs / mole], z is the number of electrons involved in the reaction and ln denotes the natural logarithm. If natural logs are replaced by base 10 logarithms, these must be multiplied by a conversion factor of 2.303). Since the numerical value of 2.303RT/F is about 60mV at 37oC we can use : E ion = 60 log 10 ( [Ion] extracellular [Ion] intracellular ) a) If [K+] extracellular = 5mM, [K+] intracellular = 148mM, [Na+] extracellular = 142mM, [Na+] intracellular = 10mM, calculate E K and E N a. b) What condition will have the more dramatic effect: an increase of 5mM of the extracellular concentration of Na+ or of K+? By how much? (Supplementary question: knowing that cardiac cells have electrical properties very similar to neurons, if you wanted to kill somebody, which one of these 2 ions would you inject him?). Exercise 6 : The Goldman-Hodgkin-Katz Equation The membrane potential of our cells is the result of the concentration gradients established for many different ions across the cell membrane (such as Na+, K+, Ca++ and Cl-). The distribution of the various ions is dependent not only on the electrochemical gradient for each ion, but also on the ability of a given ion to cross the membrane, or permeability. Therefore the permeability of the membrane for a given ion also determines the value of the membrane potential. The dependence of membrane potential on permeability and concentration is given by the Goldman-Hodgkin-Katz equation: E membrane = RT zf ln(p Ion 1 [Ion 1 ] out P Ionm [Ion m ] out )/(P Ion1 [Ion 1 ] in P Ionm [Ion m ] in ) where P Ion1...m are the permeabilities for the various ions present across the membrane, and [Ion 1...m ] in and [Ion 1...m ] out are the intracellular and extracellular concentrations of the respective ions and R, T and F are as for the Nernst equation. If we take only Na+ and K+ into account, and if Em is experimentally observed to be at -77 mv, for wich one of these 2 ions is the membrane more permeable? (use your results form exercise 5). Find the ratio P K /P Na. Exercise 7 : Basic model circuit Membranes can be modelled as electrical circuits. Each ion that the cell membrane is permeable to can be modelled as a resistor and a battery connected in series. The resistance depends on the permeability of the membrane to that given ion: the higher the permeability, the lower the resistance for that ion. The ion species also generates an electrical potential across the membrane as expressed by its Nernst potential, and is represented by a battery in the model circuit. This simple representation for each given ion can be integrated into a full equivalent circuit for the cell membrane by connecting the circuit for each ion in parallel. The full model can then be analyzed using basic circuit theory so that the voltage across the membrane and current for a particular ion can be determined. Some useful relationships for this analysis are Ohm s law and Kirchhoff s laws:

6 Ohm s Law: Kirchhoff s Current Law (KCL): Kirchhoff s Voltage Law (KVL): V=I R (voltage equals current times resistance) The sum of all currents entering and leaving any node in a circuit is equal to zero. The sum of all voltages around a closed loop is equal to zero Using the equivalent circuit representation of nerve membrane given in Fig 6 and the data listed in the following table, calculate the unknown factors. Assume steady state conditions (ie sum of currents is zero). REMEMBER: currents and voltages have polarity (can be negative or positive)! Check the signs! Figure 6: A membrane modelled as electrical circuits (The round thing on the right is the symbol for a Voltmeter...) Given Vm=-30mV; ENa=+50mV; EK=-70mV; RNa=1MOhm; RCl=infinity ENa=+55mV; EK=-90mV; ECl=-70mV; RNa=45MOhm; RK=18MOhm; RCl=85MOhm Vm=-45mV; ENa=+60mV; EK=-85mV; RK=7MOhm; RCl=infinity Find RK Vm RNa, INa (is INa inward or outward?) Exercise 8 : The membrane capacitance From Ex 6, you know that there is typically an exess of negative charge on the inside surface of the cell membrane, and a balancing positive charge on its outside surface. In this arrangement, the cell membrane creates a capacitance C m. The voltage across the membrane V m and the exess charge Q are related by the standard equation for capacitors : C m V m = Q. 1. For a neuron with a total membrane capacitance of 1nF, how many coulombs are required to produce a resting potential of -70 mv? If a single charged ion has a charge of c, how many ions does this represent? 2. The time derivative of the above equation gives C m (dv m /dt) = dq/dt. Recall that dq/dt = I, the current passing the membrane. What current (in na) will change the membrane potential of a neuron with a capacitance of 1nF at a rate of 1mV/ms? 3. Plot Vm as a function of t, if a constant current is injected with an electrode into a cell with a membrane whose resistance is infinite (no leaking current). 4. In fact, the situation is a little more complicated, since some charges will always leak through the membrane. Sketch the circuit you use to model these characteristics, with a membrane resistance and membrane capacitance. Did you put them in serie or in parallel? 5. The timecourse of the change in voltage for a given step injection of current is described by the equation: Vm(t) = R I in (1 e t/k ), where I in is the injected current and k=r C (membrane resistance times membrane capacitance). The charging curve for an Aplysia neuron is shown in Fig 7. The cell is at its resting potential of -45 mv in the beginning. Then a constant current of 9.2 na was injected. An exponential function can be fitted to that curve. This yields that the time constant (to reach 1/e of original difference resting potential to asymptotic value) is sec, the asymptotic value of the depolarization is mv. Determine (i) the total membrane resistance (ii) the membrane capacitance. Exercise 9 : Voltage-Clamp

7 Biophysics of Neural Computation, Introduction to Neuroinformatics (Winter 2006/2007) Exercise 4 Question 1 Figure 7: Charging curve of a neuron when a constant current is injected When a normal, healthy squid axon is voltage-clamped in artificial sea water, one obtains the following membrane current record in response to a step change in membrane potential from Vm= -70 mv to Vm= 0 mv. Figure 8: Voltage clamp experiment Draw similar plots of Im vs. t (when Vm is stepped from 70 mv to 0 mv) when the recordings are made under each of the following experimental conditions. For each of your plots, explain in one or two sentences how and why your graph differs from that drawn above. a) Tetrodotoxin (that blocks Na + channels) is added to the bath surrounding the axon. b) TEA (that blocks K + channels)is added to the interior of the axon. c) [Na + ]out is adjusted so that [Na + ]out =[Na + ]in d) [K + ]out is adjusted so that [K + ]out [K + ]in e) Ouabain, a specific inhibitor of the Na + -K + pump, is added to the bath five minutes before the experiment. When a normal, healthy squid axon is voltage-clamped in artificial sea water, one obtains the membrane current record shown in Fig 8, in response to a step change in membrane potential from V m = -70 mv to V m = 0 mv. Question 2 Draw similar plots of I m vs. t (when V m is stepped from -70 mv to 0 mv) when the recordings are made under Here is a momentary I-V relation of the squid giant axon ( Ii=INa+IK ): each of the following experimental conditions. For each of your plots, explain in one or two sentences how and why your graph differs from that drawn in Fig Tetrodotoxin (that blocks Na+ channels) is added to the bath surrounding the axon. 2. TEA (that blocks K+ channels)is added to the interior of the axon. 3. [Na+]out is adjusted so that [Na+]out =[Na+]in 4. [K+]out is adjusted so that [K+]out [K+]in 5. Ouabain, a specific inhibitor of the Na+-K+ pump, is added to the bath five minutes before the experiment. Exercise 10 : I-V curve In Fig 9 you see a momentary I-V relation of the squid giant axon ( I i =I Na +I K ): 1. Why can you tell that on this graph the K conductance (g K ) doesn t depend on the voltage? 2. Draw I Na, I K, I i if g K is 2 times bigger. 3. Draw I Na, I K, I i if g Na also doesn t depend on the voltage. Exercise 11 : Deriving Hodgkin-Huxley s model This is the Hodgkin-Huxley model for generation of an action potential:

8 d) [K + ]out is adjusted so that [K + ]out [K + ]in e) Ouabain, a specific inhibitor of the Na + -K + pump, is added to the bath five minutes before the experiment. Question 2 Here is a momentary I-V relation of the squid giant axon ( Ii=INa+IK ): Figure 9: Voltage clamp experiment I ionic = g L (V m V L ) + g KMAX n 4 (V m V K ) + g NaMAX m 3 h(v m V Na ) You know now that the membrane of a neuron has a negative potential at rest, that results from ion flow. The current for each ion i depends on the conductance of the membrane, and on «how far» from the resting potential the membrane potential is: I ion = g ion (V m V ion ) For several ions, the conductance is not constant, and can thus be written in term of a maximal conductance multiplied by a «gate variable»: g ion = n ion g ionmax a) What are the lower and upper bound for n? We now only consider the ions flow through channels (the constant current are typically grouped into a single term called the leakage current, the first term in the model). Imagine that in each channel their is a gate, that can be open or closed with a certain probability. In this case, since n ion represents the proportion of the maximum conductance (if all channels were open), n can be seen as the probability for one of these gates to be open. Suppose now that each channel has several gates in serie (let s say x gates), that have all to be open to let the specific ions to go through the channel. b) If the probability of a single gate to be open is n, what is the probability that all the x gates are open? The processus is dynamic, so each gate can move from the permissive to the non-permissive state and vice-versa c) If the probability that a gate in a closed state becomes open is A, and the probability that an open gate closes itself is B, how does the proportion of open gates n vary? dn/dt =... d) Do you think A and B should be voltage dependent? Hodgkin & Huxley found that a model with 4 gates in series fits well the S-shaped curve of I K during the axon potential. e) Explain the second term of the model So far for Potassium. But Sodium channels open only transiently when the membrane potential is depolarized. Therefore H&H introduced an other variable that can be seen as a ball, sometimes blocking an open channel. The probability that an open channel is not blocked is called h. f) Can you explain the last term of the model? g) Draw schematically the evolution of m, h and n as function of time during an action potential. Exercise 12 : Circuit elements Match these cellular elements to their equivalent circuit elements: Resistor Capacitor Battery Voltage source Lipid bilayer Ionic concentration gradient Ion channel Transmembrane voltage

9 Exercise 13 : Back to basics Draw an action potential on a voltage-time graph. On the t-axis indicate where the Na and K channels open and close. On the V-axis represent E K, E Na, E m and θ (the threshold). Exercise 14 : Historical figure Biophysics of Neural Computation, Introduction to Neuroinformatics (Winter 2006/2007) Exercise 6 In Fig 10 you see the original figure from the Hodgkin & Huxley (1952) paper. On the vertical axis n (running Question 1 (Potassium current according to the Hodgkin & Huxley model) from 0 to 1) is plotted, on the horizontal axis the membrane voltage. Unlike today Hodgkin & Huxley referred the voltage tobelow the you resting see thepotential original figure andfrom withe opposing Hodgkin & Huxley sign as (1952) compared paper. Onto thetoday s vertical axis convention. n infinity (running from 0 to 1) is plotted on the horizontal axis the membrane voltage. Unlike today Hodgkin & Huxley referred the voltage to the resting potential and with opposing sign as compared to today s convention. Using their data, Using calculate their data, calculate the steady-state the steady-state K+ K + current (I (I K!) K after ) after the axon theisaxon stepped is from stepped resting from potential resting potential (-60 mv) to (-60 0 mv. mv) gto KMAX 0 mv. g K is max is given to be ms/cm2, 2, and theand equilibrium the equilibrium potential of K + potential is 72 mv. All of voltages K+ is -72 mv. All voltages in the in the text text are according to the tomodern the modern convention, convention, i.e. referred toi.e. the extra-cellular referred tospace. the extra-cellular space. Figure 10: Potassium current according to the Hodgkin-Huxley model Question 2 (Cable equation) The membrane potential V(x,t) is determined by solving the following partial differential equation (cable equation) : tau* (!v/!t) = lambda 2 * (! 2 v/!x 2 ) - v + r m*i e, where tau = (c m*r m) sets the scale for the temporal variation in the membrane potential lambda = [(a*r m)/(2*r L)] 1/2 sets the scale for the spatial variation a = radius of the axon (= 2 microns) v = V-V rest r m = specific membrane resistance (= 1 MOhm mm 2 ) r L = longitudinal resistance (= 1kOhm mm) i e = the current injected into a cell