How To Solve The Cable Equation

Transcription

1 Cable properties of neurons Eric D. Young Reading: Johnston and Wu, Chapt. 4 and Chapt. 13 pp Neurons are not a single compartment! The figures below show two snapshots of the membrane potential in a model of the dendritic tree of a Purkinje cell from the cerebellum during a dendritic action potential. Note the substantial differences in potential across the dendrites and also how potential spreads through the tree with time. (from De Schutter and Smolen, 1

2 Synapses do not distribute randomly over the surface of a neuron. For example, inhibitory synapses are often located on the soma and proximal dendrites, whereas excitatory synapses are located further out on the dendrites. The examples at right show distribution of synaptic terminals of various types (identified by color) on the somas and proximal dendrites of neurons in the dorsal cochlear nucleus (distal dendrites were not reconstructed). Fusiform cells are principal cells and the other two are inhibitory interneurons. Their connections are shown. Rubio, 2004 The questions for this and the next lecture: 1. What difference does it make where a synaptic terminal is located? Is there a difference between these two terminals because of their locations, for example? 2. How do synapses at different locations interact? Do they interact more strongly if they are close together? Is there a difference between the interactions of synapses located on the same versus different dendritic branches? A component of this question is to explain why inhibitory synapses cluster near the soma. 2

3 Introduzione ai tessuti eccitabili: il neurone 30 m Neuroni dell ippocampo Simbolo

4 Distributed properties of dendritic trees are analyzed using the cable model Δx I i (x-δx) I i (x) #1 #2 #3 I i (x+δx) I m (x-δx) Δx I m (x) Δx I m (x+δx) Δx Ion channels in the membrane go here V i (x-δx) I i (x-δx) V i (x) I i (x) I V i (x+δx) i (x+δx) I m (x) Δx r i Δx #1 #2 c m Δx #3 r e Δx V e (x-δx) I e (x-δx) V e (x) I e (x) V e (x+δx) I e (x+δx) The full derivation of the cable model is given in the notes attached to this lecture. A brief summary is given below. Applying Kirchoff s current law at one of the nodes, I m (x) = I cap (x) + I ion (x) There are factors of Δx multiplying each term in the model, because the size of the current is proportional to the length of the compartment. These cancel. The membrane current I m (x) is the difference between the current I i (x-δx) flowing into the node and the current I i (x) flowing out of the node. This difference is proportional to the spatial derivative of I i, - I i / x. In turn, I i (x) is proportional to the spatial derivative of membrane potential -(1/r i ) V i / x, which is Ohm s law, so the membrane current is the divergence of membrane potential (1 r i ) 2 V i x. 2 1 r i 2 V i x 2 1 r e 2 V e x 2 3

5 By the same argument at the extracellular node, 1 r e 2 V e x 2 = I m = 1 r i 2 V i x 2 Now the transmembrane potential V = V i - V e so 2 V x = 2 V i 2 x 2 V e = (r 2 x 2 i + r e )I m so that 1 2 V r i + r e x = I 2 m = I cap + I ion The current through the capacitor is I cap (x)δx = c m Δx V i most general form of the cable equation. 1 2 V r i + r e x = c V 2 m t + I ion t, which gives the 1 r i 2 V i x 2 1 r e 2 V e x 2 To obtain analytical solutions, the membrane is linearized Now the ionic current is given by I ion =g m (V-E rest ). Because the differential equation is now linear it is possible to change the reference point for the membrane potential from 0 mv across the membrane to the resting potential. This amounts to a change in variable in the differential equation, replacing V-E rest with V. This gives the linear cable equation: 1 r i + r e 2 V x 2 = c m V t + g mv Which is usually written as below, defining a length parameter λ and a membrane time constant τ m. 2 V λ 2 x = τ V 2 t + V where λ= 1 g m (r i + r e ) and τ m = c m g m 4

6 Parameters: it is useful to relate the parameters of the ladder model to properties of the cylinder membrane. c m = cap. per unit length of cylinder = 2πaC g m = conductance of unit length of cyl. = 2πa /R m r i = resistance of unit length of cytoplasm = R i /πa 2 where C is the capacitance of a unit area of membrane, 1 µfd/cm 2 R m is the resistance of a unit area of membrane, Ω. cm 2 R i is the resistance of a unit cube of cytoplasm, and a is the radius of the cylinder. 200 Ω. cm THEN the two parameters of the cable equation are given by λ = 1 g m (r i + r e ) 1 = a R m and τ = c m = R m C g m r i 2R i g m Electrotonic processing in dendrites: potentials become smaller in amplitude and more spread out in time as they propagate away from the source 0.1τ I(t) Membrane potential V Distance down the cylinder, in units of λ V(x,t) Time, in units of τ 2 V λ 2 x = τ V 2 t + V V(x,t = 0) = 0 V x x=0 = I(t) r i and V(x,t) = finite x,t Jack, Noble, & Tsien,

7 Parameters of the cable model: λ is the length constant. To see why, consider the steady-state distribution of membrane potential, say in response to a steady current after a long time. In this situation, V/ t = 0 and the cable equation becomes λ 2 d 2 V dx 2 V = 0 The homgeneous solution takes the form V (x) = Ae x / λ + Be x / λ where A and B are constants determined from the initial conditions. The solutions vary exponentially with distance x divided by λ, showing that λ determines the distance through which disturbances spread along a cable. λ = 1 g m r i = R m 2R i a Note that the length constant is proportional to a 1/2, so potentials spread further in larger cylinders. For a 1 µm dendrite with R m =3x10 4 Ω. cm 2, λ = 866 µm Parameters of the cable model: τ is the time constant. In all solutions to the cable equation, time appears as t/τ, so that τ determines the time scale of solutions, e.g. how long it takes membrane potential to change in response to an injected current. τ = c m g m = R m C Note that the time constant does not depend on the cylinder size. For a process with R m =3x10 4 Ω. cm 2, τ = 30 ms. This decay time is approximately τ. 6

8 The cable model at work: EPSCs recorded in the soma show the effects expected, depending on the dendritic source (smaller and slower if initiated further away) soma stimulation sites on dendrites Bekkars and Stevens, 1996 The distance that a potential propagates in a membrane cylinder is set by the length constant λ. For this reason, a meaningful measure of the length of a membrane cylinder is its electrotonic length, equal to physical length divided by λ. For a semi-infinite cylinder, the potential decays as e -x/λ with distance. For finite cylinders, the decay depends on the boundary condition (or load) at the second end. V-clamped open circuit I(t) V-clamped short circuit L= V(x)? Load X = x / λ Electrotonic length physical length (mm) = L = λ 7

9 How large is a cell? This can be answered in terms of physical length, as in the picture at right. A more meaningful answer is in terms of electrical size, some measure of the electrical coupling between two points. Two measures: 1. Electrotonic length (as defined previously) P 2. The morphoelectrotonic transform (MET), in which distance is defined in terms of voltage attenuation A as follows: then the MET is L PQ = l PQ λ A PR = V R V P resulting potential observed at point R membrane potential produced at point P, say by a synapse R l PQ Q Λ PR = ln A PR Note that Λ PR incorporates both the exponential decay of potential with distance and the effects of branching (which the cumulative electrotonic length L PQ +L QR would not). Justification for the MET: Consider again a semi-infinite cylinder driven by a constant current I 0 at one end, at times where the potential is in steady state. I 0 V(x)=V 0 e-x/λ The voltage gain from the point of current injection to point x is and the MET is A(x) = V (x) V 0 = e x / λ Λ(x) = ln( e x / λ ) = x λ = L(x) so in this special case, the MET corresponds to to the electrotonic length. Again, the advantage of the MET is that it takes into account the effects of branching, not relevant in this example. 8

10 How large is the dendritic tree? NOTE that the MET is different depending on the direction in which it is defined. physical size electrotonic length, equal to length/length const C) Λ IS D) Λ SI MET, dendrite to soma Note smaller! MET, soma to dendrites (note scale!) Zador, 1993 A cell s electrical size depends on the amount of synaptic input it receives. The somaward METs at right are for a cell with no synaptic input (left) and a cell with substantial, randomly occurring, input (right). Note the cell is electrically larger with synaptic input. This is explained as an effect of synaptic input on R m and therefore on λ, since λ = R m 2R i a (λ decreases as R m decreases, making the cell electrically larger.) Bernander et al.,

11 Course Notes: Cable Theory Additional reading: Koch and Segev, chapt 2, 3, and 5; Johnston and Wu, Chapt. 4. Cable theory was originally applied to the conduction of potentials in an axon by Hodgkin and Rushton (1946) and was later applied to the dendritic trees of neurons by Rall (1962a). The theory itself is much older and was first developed for analyzing underwater telegraph transmission cables. The general problem addressed by cable theory is how potentials spread in a dendritic tree. This is electrotonic conduction and is assumed to occur by passive process, i.e. without action potentials. A typical neuron has thousands of synaptic inputs spread across its surfaces. Cable theory is concerned with how these inputs propagate to the soma or the axon initial segment, how these inputs interact with one another, and how the placement of an input on a dendritic tree affects its functional importance to the neuron. Derivation of the cable equation Previously, we have considered only point neuron models. That is, we have assumed that the neuron is electrically compact, so that it can be represented by a single patch of membrane. The gist of this assumption is that the membrane potential is the same everywhere in the neuron. In real neurons, this assumption is not true. Substantial differences in potential exist along the length of a neuron s processes and the resulting longitudinal currents must be explicitly considered. Figure 1 shows three stages of abstraction of a neuron s dendritic membrane. Figure 1A shows a sketch of the neuron, with its soma at right and a dendrite which branches twice spreading off to the left. One length of membrane cylinder from a secondary branch is isolated in Fig. 1B. It is assumed that the cylinder is of uniform radius along its length (although this assumption can be relaxed, Rall, 1962a). The cylinder is divided into three portions of equal length Δx along the x axis, which runs from left to right. Figure 1C shows an electrical cable model for this length of cylinder. Each of the subcylinders labeled #1, #2, and #3 is assumed to be an isopotential patch of membrane. The membrane of each subcylinder is represented by a parallel combination of membrane capacitance c m Δx and an unspecified circuit for the ionic conductances in the membrane, represented by a box. The total current through a membrane patch is I m (x)δx. Note that the membrane current varies with distance x down the cylinder. I m and c m are membrane current and capacitance per unit length of cylinder so that multiplying by Δx gives the total current and capacitance in a subcylinder. The membrane potentials inside the cell V i (x) and outside the cell V e (x) are shown at the nodes in Fig. 1C. It is assumed that the potentials also vary with distance down the cylinder, so they are functions of distance x. The membrane potential is V i (x)-v e (x) as usual. Because the potentials vary along the length of the cylinder, there will be currents I i (x) and I e (x) flowing between the nodes. I i (x) is the total current flowing down the interior of the cylinder and I e (x) is the total current flowing parallel to the cylinder in the extracellular space. In a real brain there will be many cylinders from different neurons packed together, so there will be many extracellular currents. I e (x) is only the portion of the extracullar current associated with the cylinder under study.

12 2 The internal current I i (x) flows through resistance r i Δx, which is the resistance of the solutions inside the cylinder between the center of one subcylinder and the center of the next. r e Δx is similarly defined as the resistance in the extracellular space between the center of two subcylinders, i.e. as the resistance to the flow of current I e (x). r i and r e are again defined as Figure 1. A Sketch of a portion of the dendritic tree of a neuron emerging from the soma at right. B Portion of a secondary dendrite divided into three subcylinders. The axial current I i and the membrane current I m are shown next to the arrows. C Discrete electrical model for the three subcylinders. Axial currents flow from one subcylinder to the next through resistances r i Δx. Membrane currents flow through a parallel combination of the membrane capacitance c m Δx and membrane ion channels, represented by the boxes. Explicit circuits for the boxes are shown in Fig. 2.

13 3 resistances per unit length of cylinder. Ohm s law for current flow in the intracellular and extracellular spaces gives: V i (x) V i (x + Δx) = I i (x) r i Δx and V e (x) V e (x + Δx) = I e (x) r e Δx (1) Rearranging and taking the limit as Δx goes to 0, lim Δx V i(x + Δx) V i (x) Δx = V i x = r i I i (x) and V e x = r e I e (x) (2) Conservation of current at the intracellular and extracellular nodes gives I i (x Δx) I i (x) = I m (x) Δx I e (x Δx) I e (x) = I m (x) Δx or or I i x = I m(x) I e x = I m (x) (3) Defining the membrane potential as V = V i -V e allows the membrane current I m to be written as the sum of the ionic current I ion (x,v,t) through the box and the current through the membrane capacitance: I m (x) Δx = I ion (x,v,t) Δx + c m Δx V t (4) The ionic current is in general a complex and nonlinear function of membrane potential modeled, for example, by Hodgkin-Huxley type equations. As for I m, I ion is ionic current per unit length of membrane cylinder. Differentiating and subtracting Eqns. 2 and substituting Eqns. 3 allows the following relationship between membrane potential and membrane current to be written: 2 V x 2 = 2 (V i V e ) x 2 I = r i i x + r I e e x Substituting Eqn. 4 gives the nonlinear cable equation: = (r i + r e ) I m (5) 1 2 V r i + r e x 2 = c V m t + I ion (6) Equation 6 models the distribution of membrane potential in a membrane cylinder. The right hand side is the usual equation used for a point neuron model, and expresses the fact that the total

14 4 membrane current at a point is the sum of the currents through the membrane capacitance and the ion channels. The left hand side is the current injected into the point by the rest of the system, i.e. the current which spreads from adjacent points on the cylinder. The ionic current I ion can be modeled by the usual Hodgkin-Huxley equations. Equation 6 is linear, but the model for I ion is not, if a full Hodgkin-Huxley model is used. It is useful to consider a simplified, completely linear, version of the cable equation. This is accomplished by using a linear model for I ion. Figure 2 shows circuit diagrams for the membrane I m I ion g c Na g K g L m E Na E K E L = c m g m + E rest Figure 2 Circuits for the membrane patch. At left is a full Hodgkin-Huxley model, in which g Na and g K are nonlinear conductances. At right, the resistor-battery circuits have all been combined into a single Thévenin equivalent. In the linear cable model, g m is assumed to be a constant, linear, conductance. model, i.e. for the contents of the boxes in Fig. 1. At left is the full model with sodium, potassium, and leakage conductances. In a real neuron, there could be additional parallel battery-resistor combinations for calcium conductances and for multiple kinds of sodium and potassium conductances. In the circuit at right, the battery-resistor pairs have been combined into a single equivalent battery and resistor representing the resting potential E rest and the resting membrane conductance g m. The single battery-resistor circuit at right is a Thévenin equivalent. See Question 2 for the relationship between the two circuits. The resistors in the circuit at left in Fig. 2 are non-linear and the membrane conductance g m in the circuit at right is also non-linear, in the absence of further assumptions. To derive a linear cable equation, it will be assumed that g m is a constant, linear resistance equal to the resting conductance of the membrane. This is an approximation which is valid only to the extent that membrane potential excursions are small enough not to induce significant gating of the voltagedependent channels in the real membrane circuit. Given recent evidence, it is clear that dendritic trees contain significant densities of voltage-gated channels and that these channels participate in the responses to synaptic activation. Nevertheless, there are many insights into the functioning of dendritic trees that can only be gained from analysis of the linear cable model. If g m is a linear resistance, then I ion =g m (V-E rest ). Substituting this in Eqn. 6 and changing the membrane potential variable to v=v-e rest, gives the linear cable equation.

15 5 1 2 v r i + r e x 2 = c v m t + g v mv = c m t + 1 v (7) r m where use has been made of the fact that, because E rest is a constant 2 V x 2 = 2 (V E rest ) x 2 = 2 v x 2 and V t = (V - E rest ) t = v t (8) Equation 7 can be rewritten in a non-dimensional form by multiplying both sides by membrane resistance r m r m 2 v r i + r e x 2 = r v m c m t + v or 2 v λ2 x 2 = τ v m t + v (9) Two new constants are defined here, the length constant λ and the membrane time constant τ m. These names will be justified in terms of the solutions derived in later sections. Now define new dimensionless distance and time variables χ and T as χ = x / λ and T = t / τ m. Finally, the linear cable equation can be written in terms of the dimensionless variables as 2 v χ 2 = v T + v (10) which follows from the chain rule for derivatives. Equation 10 is the form in which the cable equation will be used for most of the rest of the discussion. Question 1. Show that V e 2 x 2 = re I m. Under appropriate conditions, this equation can be used to infer membrane current density I m from extracellular potential measurements V e only. What anatomical features of the system are necessary to allow this calculation? Hint: the equation is one-dimensional; under what conditions is the extracellular potential near a group of neurons likely to vary along one axis only? Describe how the calculation would be done. Question 2. Show that, for the circuits in Fig. 2: E rest = g NaE Na + g K E K + g L E L g Na + g K + g L and g m = g Na + g K + g L (11) Question 3. Consider the case in which the membrane consists only of a leakage channel and a Hodgkin-Huxley type delayed rectifier potassium channel (left side of Fig. 3). This nonlinear circuit can be approximated by the linear circuit at right in Fig. 3; the approximation is accurate if membrane potential excursions from the resting potential are small. Derive a linear cable equation for this small-signal equivalent model for the membrane patch. Note that it will not be possible to write this as a single equation in the form of Eqn. 7; instead two linear differential equations will be required. However if the system is Laplace or Fourier transformed, then a single equation of the form d 2 V dx 2 = AV can be derived. A is a complex scalar function of the Fourier or Laplace

16 6 transform variable and V is the Fourier or Laplace transformed membrane potential. Derive an expression for A in terms of the parameters of the linear circuit in Fig. 3. Figure 3 At left is a circuit patch model for a membrane containing only a leakage channel g L and a Hodgkin-Huxley type delayed rectifier channel g K. At right is a linear approximation of the nonlinear circuit. Cable equation parameters The parameters of the cable model are defined in this section. Various electrical parameters of neural cables are introduced and defined. For all the following, a is the radius of the membrane cylinder. r i = resistance of the solution inside a unit length of membrane cylinder to axial current flow I i, with units like Ω/cm. = R i πa 2 where R i is the resistivity of the solution, defined as the resistance from one face to the opposite face for a cube of solution of unit dimensions. Typically R i is Ω. cm for neural cytoplasm. r i is equal to the resistance of a unit length of cylinder of cross sectional area πa 2. c m = capacitance of the membrane of a unit length of membrane cylinder, with units like µfd/cm. = 2πaC where C is the capacitance of a unit area of membrane. For neurons, C 1 µfd/cm 2. c m is the capacitance of a unit length of cylinder with radius a, which therefore has surface area 2πa. g m = 1/r m = conductance of the membrane of a unit length of membrane cylinder, with units like S/cm. Note that the resistance of a unit length, r m has units Ω. cm. = 2πa / R m where R m is the resistance of a unit area of membrane. For neurons, R m is generally in the range Ω. cm 2. It would be more intuitive to express R m as a conductance with units S/cm 2 by analogy to C. However, in virtually all publications, this constant is given in terms of resistance. g m or 1/r m is the conductance of a unit length of cylinder with surface area 2πa.

17 7 r e = resistance to longitudinal flow of current in the extracellular space, with units like Ω/cm. This constant is not defined in terms of the parameters of the membrane cylinder. Instead, it is assumed to be negligible compared to the other impedances in the circuit and ignored, i.e. r e = 0. Given the parameters defined above, the length constant and membrane time constants can be specified in terms of more fundamental parameters. These are given below along with a new constant G which is the input conductance of an semi-infinite cylinder. The properties of G will be derived in a later section. λ = length constant, with units cm. From Eqn. 9, = r m r e + r i r m r i = R m a 2 R i (12) where the assumption r e <<r i has been used to eliminate r e from the equation. Note that λ varies as the square root of cylinder radius a. τ m = membrane time constant. Also from Eqn. 9, = r m c m = R m C Note that τ m does not depend on cylinder radius. G = input conductance of a semi-infinite cylinder at its end. = 1 r i λ = 2 πa 3/ 2 (13) R i R m The dependence of G on the 3/2 power of cylinder radius will be important in considering the equivalent cylinder theorem in a later section. Solutions for a semi-infinite cylinder The basic properties of electrotonic conduction can be seen by considering a simple case, a semi-infinite cylinder driven with a step of current at its end. The cylinder is shown in Fig. 4. The step of current is injected at the end of the cylinder (x=χ=0) and we want to know the time course of membrane potential in the cylinder at various positions. I 0 u(t) x=0 χ=0 I i (t)... Figure 4. Semi-infinite cable driven by a step current at its end.

18 8 The problem to be solved is the cable equation, Eqn. 10, with suitable boundary conditions. The cable equation needs one boundary condition in the time domain and two in the spatial domain. For almost all cable theory problems, the appropriate time boundary condition is a zero initial condition, i.e. v(χ,t=0) = 0. One spatial boundary condition for this case is that the axial current in the cable must equal the external current injected. Using Eqn. 2 and assuming that r e =0 so that V e can be ignored, the boundary condition can be written in terms of membrane potential as I 0 u(t) = I i (χ = 0,T) = 1 v = 1 v v = G (14) r i x x=0 r i λ χ χ=0 χ χ=0 Note that v, the membrane potential relative to rest, has been substituted for V, the absolute membrane potential, as justified by Eqn. 8. Eqn. 14 provides one boundary condition in space. For this problem, a second boundary condition cannot be given, but the condition that the membrane potential must remain finite over the whole cable (a regularity condition) will suffice as a second constraint. The problem to be solved is as follows: 2 v χ 2 = v T + v v(χ,t = 0) = 0 (15) v χ χ=0 = I 0 G u(t) and v(χ,t) < for all χ,t A convenient way to solve problems of this kind is to Laplace transform the variables over the time domain. The Laplace transform of a time function v(t) is defined as follows: V (s) = L[ v(t) ] = v(t)e st dt (16) 0 where V (s) is the transformed function and s is the transform variable. Laplace transforms are useful because of the following property: L[ dv dt] = sv v(t = 0) (17) With this property, differential equations are reduced to algebraic equations, which often simplifies solving the equations. gives Transforming the cable equation problem of Eqn. 15 along with its boundary counditions

19 9 2 V χ 2 = (s +1)V V χ χ =0 = I 0 sg and v(χ,t) < for all χ,t (18) The membrane potential variable v(χ,t) has been replaced with the transformed variable V (χ,s). Because Laplace transformation and differentiation are both linear operations, the Laplace transformation does not affect the derivatives with respect to χ in Eqn. 18. The time derivative has been replaced with a multiplication by s as in Eqn. 17 and the zero initial condition has been used as part of that operation. The spatial boundary condition at 0 was transformed, using the fact that L[ u(t) ] = 1 s. The spatial regularity condition of finite v cannot be transformed directly, but will be used below. Equation 18 is now an ordinary differential equation whose solution takes the form V (χ,s) = A(s)e s+1 χ + B(s)e s+1χ (19) where A(s) and B(s) are to be determined from the boundary conditions. Using the boundary condition at 0, V = s +1 A(s) e s+1 χ s +1 B(s) e s+1 χ χ χ=0 [ ] χ=0 = s + 1 [ A(s) B(s) ] = I 0 sg (20) The additional condition needed to uniquely specify A and B is the regularity condition. From experience with exponential solutions like Eqn. 19, it seems likely that either A or B should be zero so that as χ, the membrane potential remains finite. However, the solution in Eqn. 19 is in terms of the Laplace transform and it is not clear how to directly apply this condition. To apply the regularity condition, assume that B(s)=0; in that case Eqns. 19 and 20 give the following solution: V (χ,s) = I 0 G e s+1 χ s s +1 (21) The regularity condition must hold in all conditions, including in the steady state as T. This so-called final value can be computed from a theorem of Laplace transforms: lim v(t) = lim sv (s) t s 0 (22) Applying the final value theorem to Eqn. 21 gives v(χ,t ) = I 0 e χ G. Clearly this function does not remain finite as χ goes to infinity. Thus we must conclude that A(s)=0 and the solution for the Laplace transform of membrane potential is

20 .. 10 V (χ,s) = I 0 e s+1 χ G s s +1 (23) The membrane potential function can now be obtained by inverting the Laplace transform in Eqn. 23. The method for accomplishing this is described in Jack et al. (1975, chapter 3) and in Hodgkin and Rushton (1946). The solution is as follows: v(χ,t) = I 0 2G e χ χ erfc 2 T T e χ χ erfc 2 T + T (24) The function erfc(x) is the complementary error function, defined as x erfc(x)= 1 2 e ξ2 dξ π (25) 0 Figure 5A shows plots of exp(-x 2 ) and erfc(x) for reference; erfc is a standard function and algorithms for computing it are found in Matlab and other mathematics programs. A erfc(x) 2 1 e -x2 v(χ,t) B G I T= v(χ,t) G I 0 C x 1 χ=0 (1-e -T ) χ=1 0.2 χ=2 χ= T D w= v(χ,t) v(χ, ) χ χ=0 χ=1 χ=2 χ=3 χ= T Figure 5. A. Plots of exp(-x 2 ) and erfc(x). B. Plot of the distribution of the membrane potential along the cylinder (Eqn. 24) at 6 times: T=0.01, 0.1, 0.2, 0.4, 1.0, and. Membrane potential is normalized by its steady state value at χ=0 (I 0 /G ). C. Plot of the time course of membrane potential at four locations along the cylinder, normalized as in B. For comparison, an exponential rise is also plotted. D. Same as C except membrane potential is normalized by its steady state value. Several important properties of the solutions to the cable equation can be seen from plots of Eqn. 24 in Fig. 5.

21 11 1. Figure 5C shows plots of the growth of membrane potential in response to the step current injection; each curve shows potential growth at a different position along the cylinder, as labeled. Note that at χ=0 the potential grows to a steady value relatively quickly. The growth is faster than exponential, as seen by the comparison with the function (1-exp(-T)), which is plotted for comparison. As the point moves away from the end of the cable (larger χ), the growth of potential is delayed and the steady state value of potential is smaller. The difference in growth rate can be seen in Fig. 5D where the same plots are shown, except normalized by their maximum value. Note that the membrane time constant τ m sets the time scale of the response, in that the abscissae of Figs. 5C and 5D are scaled in units of τ m. Figure 5C illustrates the basic features of electrotonic conduction: as the potential spreads from the site of a disturbance (in this case the current injection at the end of the cable), the amplitude of the potential gets smaller and its time course is extended. In this case, the rise of the potential is delayed and its rise is slower. 2. Figure 5B shows the potential spread along the cylinder at various times following the onset of the current at T=0. The most illuminating case is for T=, i.e. in the steady state. In the steady state, Eqn. 24 becomes v(χ,t ) = I 0 G e χ = I 0 G e x /λ (26) This equation illustrates the meaning of the space constant λ. The decay of potential is exponential along the cylinder, so that potential decays by 1/e for every distance λ. Thus the space constant is a measure of how far a disturbance spreads away from the point of current injection. Equation 26 also allows the meaning of the parameter G to be understood. Note that at χ=0, v(t= )=I 0 /G. Thus the resistance looking into the end of the semi-infinite cylinder in the steady state, for application of D.C. current, is 1/G. This is the basis for the statement made earlier that G is the input conductance of a semi-infinite cable. 3. A measure of the speed of electrotonic spread can be gotten from the points marked by black circles in Fig. 5D. These circles mark the times at which the potential is half its steady state value, for different values of χ. If the χ values are plotted against the half-times, the result is a straight line with slope 2λ/τ m (Jack et al., 1975). This value can be though of as the speed of spread of electrotonic disturbances. Of course, it is not a true propagation speed, in the sense of the action potential propagation speed, because there is no fixed waveshape that is propagating, i.e. this is not a true wave. Nevertheless, this speed provides a way to calculate the time delays expected in electrotonic conduction. Note that it varies as the square root of the cylinder radius, since speed = 2 λ τ m = 2 a R m R i C 2 (27)

22 12 Question 4 Suppose that the current injected into the semi-infinite cable in Fig. 4 is an impulse, Q 0 δ(t). Solve for the potential distribution in the cable as a function of time and χ. The following Laplace transform pair will be helpful: e x s s e x2 /4t πt It will also be helpful to know the frequency shift property: L[ e at f (t)] = F (s + a). This was one model that was considered for neurons in early papers. The idea was that the dendritic tree is very long, so it could be approximated as infinite, and the soma was unimportant. Show that the membrane time constant τ m can be determined from a plot of ln t v(0,t) [ ] versus t. References Hodgkin, A.L. and Rushton, W.A.H. The electrical constants of a crustacean nerve fibre. Proc. Roy. Soc. B-133, (1946). Johnston, D. and Wu, S.M. Foundations of Cellular Neurophysiology. MIT Press, Cambridge (1995). Koch, C. and Segev, I. Methods in Neuronal Modeling from Ions to Networks (2 nd Ed.). MIT Press, Cambridge (1998). Rall, W. Theory and physiological properties of dendrites. Ann. N.Y. Acad. Sci. 96: (1962a).

23 Non-linear cables in neurons Eric D. Young Linear cable theory ignores the presence of voltage-gated channels in the membrane. Potentials become smaller in amplitude and more spread out in time as they propagate away from the source, called electrotonic or passive conduction. 0.1τ I(t) Membrane potential V Distance down the cylinder, in units of λ V(x,t) λ 2 2 V x 2 = τ V t + V Time, in units of τ Jack, Noble, & Tsien,

24 Cable theory was originally developed (by Hodgkin and Rushton in the 50 s) to apply to unmyelinated axons. In this case the cable is clearly non-linear. inside outside, assumed ground (r e = 0) HH model for the membrane The cable equation must includes the non-linearities in the transmembrane ion current term: 1 2 V r i x = c V 2 m t + I = c V ionic m t +G m 3 Na h(v E Na ) +G K n 4 (V E K ) +G m (V E rest ) plus additional differential equations to describe the evolution of m, h, and n. An important test of the HH formulation is whether it can predict the propagation of the AP along an axon. =I Na +I K +I m Koch, 1999 The action potential in axons moves at a constant velocity like a wave. The propagation of the AP depends on the spread of current away from the site of the current AP. So that the action potential repeats itself at successive locations along the axon AP direction refractory active site depolarization, next active site Koch, 1999 and Hille,

25 AP direction G K increased G K, shunts return current, refractory membrane increased G Na, inward current driven by E na depolarizing this and adjacent membrane. resting membrane, depolarized by return current Does the HH model predict the propagation of the action potential? An important attribute is that propagation is faster for larger cables. In fact, the HH equation predicts this behavior. To see how, first isolate the dependence of terms on the radius a of the membrane cylinder. 1 2 V r i x = c V 2 m t +G Nam 3 h(v E Na ) +G K n 4 (V E K ) +G m (V E rest ) Substitute for the constants that vary with cylinder radius and rewrite membrane currents as current/area of membrane membrane current as current/length of axon π a 2 2 V V = 2π ac x 2 t + 2π a [ G ˆ Na m 3 h(v E Na ) + G ˆ K n 4 (V E K ) + G ˆ m (V E rest )] R i membrane current as current/area of membrane So that finally the effects of cylinder radius can be isolated in one term: a 2 V 2R i x = C V 2 t + G ˆ Na m 3 h(v E Na ) + G ˆ K n 4 (V E K ) + G ˆ m (V E rest ) 3

26 H&H were unable to directly compute solutions to the non-linear cable equation. Instead, they argued that if the AP is to propagate without change in shape, then it must be described as a wave, as V(x,t) = F(x ut) where u is the propagation velocity of the AP. With this assumption and the chain rule 2 V x = 1 2 V 2 u 2 t 2 so that the non-linear cable equation can be written as an ordinary differential equation a d 2 V 2R i C u 2 dt = dv 2 dt + H(V,t) The HH currents have been gathered up into the term H(V,t), which does not vary with the radius of the axons. This equation could be solved by H&H (by hand). By trial and error, they found a value of the constants multiplying the leading term which gives a stable, propagating solution resembling an AP. An important test of the theory is provided by two aspects of the constants: 1. The value of the constant found by HH predicted that u = AP propagation velocity = 18.8 m/s. The experimental value in squid giant axon was 21.2 m/s. Close! 2. If a/(2r i Cu 2 ) = constant, then it follows that the propagation velocity u in an axon should be proportional to the square root of the radius of the axon. This prediction has been found to hold experimentally (?). a d 2 V 2R i C u 2 dt = dv 2 dt + H(V,t) (constant) 4

27 Axons that travel any distance in the brain are myelinated. This means that glial cells form an insulating layer around that axon by wrapping their membranes around the axon. At intervals the membrane of the axon is exposed at nodes of Ranvier. The sodium and potassium channels of these axons are concentrated at the nodes. Thus active currents associated with the action potential occur only at nodes, and the action potential jumps from node to node. Node Potassium channels Sodium channels Jacobson, 1970; Levitan & Kaczmarek, Myelinization changes AP propagation from a continuous process, as in the HH axon, to a discrete process in which the AP jumps from node to node. Propagation through the internodes is described by the cable equation, with nodal currents described by a HH-like model. Models of myelinated axons consist of patches of HH membrane, interspersed with cable models of the internodes. node of Ranvier myelin Koch,

28 The advantage of myelin is that conduction velocity is now proportional to axon radius, not the square root of radius. This result is predicted by the equations for propagation of current through the model in the previous slide. Of course, axons with velocity proportional to a instead of a are better for the brain, in that signals can be transmitted more quickly with less hardware (smaller axons) and with less energy. Excit. here What is the effect of relative placement of synapses on the dendrites? Because cells are not electrically compact, the relative placement of synapses on dendrites matters. or Why inhibitory synapses cluster near the soma. E F = E E I at 2 at S at 1 E E E I at 3, 4, 5 Koch et al.,

29 Dendritic trees are not passive: action potentials invade the dendritic tree from the soma, but not vice-versa (this is explained on the basis of the asymmetry in the MET). Note that the AP begins first in the soma even if the stimulus is in the dendrite! Stuart and Sakmann, 1994 What is the effect of relative placement of synapses on the dendritic tree? The answer depends on the properties of the cell and the type of synapse. 100 synapses were scattered on the dendrites of a model* of the cortical pyramidal cell at lower left. They were arranged in 100/k clusters of k synapses each. The synapses were then activated with independent 100 Hz spike trains and the postsynaptic firing rate determined in simulations. The higher the firing rate, the more effective is a particular distribution of synapses. for a passive tree, it is best to spread the synapses out, so they can sum linearly voltage-dependent (NMDA) synapses and active channels in the dendrites make clustering valuable Cluster Size Cluster Size Cluster Size * (the model was a direct compartmental model of the neuron shown above. Mel,

30 Neurons often are covered in spines, small extensions of dendrites on which excitatory synapses are made. Inhibitory synapses tend to occur on dendritic shafts. 0.5 µm Wilson et al.,, 1992 Typically, spines are µm in diameter, and µm long What is the effect of spines on input/output processing in a neuron? Spines do not have a significant electrical effect: the worst-case electrotonic length (L) of the spine neck is about Calculations show that the current injected into a dendrite by a synapse on a spine head is about the same as if the synapse were directly on the dendrite. V spine V d 8

31 spine In fact, spines are calcium traps, the length constant for calcium diffusion in dendrites is very short, approximately the length of a spine neck. a. shows 2-photon images of Ca in a spine and dendrite (right) and the Ca difference signal following synaptic stimulation b. Shows the Ca signals in the spine (red) and dendrite (black) for synaptic stim. c. Shows the Ca signals in spine and dendrite following antidromic AP invasion spine dendrite dendrite synaptic stim antidromic AP Yuste et al., 2000 The calcium signal in spines is an essential message for postsynaptic plasticity, discussed in the next slides. Confining Ca to a single spine makes the changes produced by that Ca specific to the synapse on the same spine. VGCaCh NMDA Ca-B Ca ER AMPA IP 3 Ca ATPase mglur Dendrite 9

32 Strength of synapses again. Synaptic strength can be modulated under behavioral conditions by metabotropic mechanisms, exemplified by postsynaptic sensitization in the aplysia gill-withdrawal reflex. The reflex protects the gill from damage using the warning signal of strong stimulation of the tail. These effects can last for up to several days, if the stimulus is repeated enough times. Phosphorylating K S channels decreases K + currents, prolonging the AP and allowing more Ca ++ to enter the presynaptic terminal. Longer-term changes in the strength of a synape occur due to use of the synapse. Below are an example of long-term potentiation (LTP, left) and long-term depression (LTD, right). The stimulus protocol involves two components: 1. Stimulation of presynaptic fibers (s) 2. Depolarization of the postsynaptic cell through the recording electrode (r) HFS LFS LTD LTP r s Amplitude of the post-synaptic current produced by stimulation of synaptic inputs. Fujino and Oertel,

33 LTP and LTD can both occur at the same synapse, as in the example on the previous slide. The difference seems to depend on the strength of the Ca signal in the postsynaptic terminal. The sequence of events occurring in the postsynaptic cell is known partially and is described below. Glutamate NMDA Ca ++ Protein receptor increase kinase (requires both glu and weaker Depolarization depolarization) EPSC Presumably the net effect depends on the relative activation of these two processes stronger EPSC Phosphatase The actual mechanisms of strengthening the postsynaptic EPSC probably include several; 1) changes in ion currents due to phosphorylation; 2) the number of AMPA receptors in the postsynaptic terminal can change; 3)?? Johnston et al

34 Computational Neural Modeling and Neuroengineering The Hodgkin-Huxley Model for Action Potential Generation

35 Action Potential Propagation in Dendrites

36 Stochastic influences on dendritic computation

37 The Hodgkin-Huxley Model of Action Potential Generation

38 Motivations Action Potentials (A) Giant squid axon at 16 C (B) Axonal spike from the node of Ranvier in a myelinated frog fiber at 22 C (C) Cat visual cortex at 37 C (D) Sheep heart Purkinje fiber at 10 C (E) Patch-clamp recording from a rabbit retinal ganglion cell at 37 C (F) Layer 5 pyramidal cell in the rate at room temperatures, simulataneuous recordings from the soma and apical trunk (G) A complex spike consisting of several large EPSPs superimposed on a slow dendritic calcium spike and several fast somatic spikes from a Purkinje cell body in the rat cerebellum at 36 C (H) Layer 5 pyramidal cell in the rat at room temperature - three dendritic voltage traces in response to three current steps of different amplitudes reveal the all-ornone character of this slow event. Notice the fast superimposed spikes. (I) Cell body of a projection neuron in the antennal lobe of the locust at 23 C

39 Historical Background Bernstein The membrane breakdown hypothesis Prior to 1940, the excitability of neurons was only known via extracellular electrodes A major mystery was the underlying mechanism By the turn of the 20th century it was known that 1) cell membranes separated solutions of different ionic concentrations 2) [K + ] o << [K + ] i 3) [Na + ] o >> [Na + ] i In 1902, Bernstein, reasoning that the cell membrane was semi-permeable to K + and should have a V m ~ -75mV, proposed that neuronal activity (measured extracellularly) represented a breakdown of the cell membrane resistance to ionic flow and the resulting redistribution of ions would lead from -75mV to 0mV transmembrane potential (V m =0)

40 Historical Background Cole et al. The space clamp Marmont (1949) and Cole (1949) developed the space clamp technique to maintain a uniform spatial distribution of V m over a region of the cell where one tried to record currents This was accomplished by threading the squid axon with silver wires to provide a very low axial resistance and hence eliminating longitudinal voltage gradients The voltage clamp Cole and colleagues developed a method for maintaining V m at any desired voltage level Required monitoring voltage changes, feeding it through an amplifier to then drive current into or out of the cell to dynamically maintain the voltage while recording the current required to do so Schematic of the voltage clamp apparatus for the giant squid axon (reproduced from Hille, 1992)

41 Historical Background Hodgkin and Katz The sodium hypothesis Hodgkin and Katz (1949) had demonstrated that both sodium and potassium make significant contributions to the ionic current underlying the action potential First to realize that, in contrast to Bernstein s theory of increased permeability for all ions, the overshoot and undershoot of the AP could be explained by bounded changes in the permeabilites for a few different ions Hodgkin and Katz postulated that during the upstroke of the AP, Na + was the most permeable ion and so the voltage of V m moved towards its Nernst potential of ~ 60mV. They predicted and then demonstrated that the AP amplitude would therefore depend critically on the external concentration of Na +. They generalized the Nernst equation to predict the steady-state V m for the case of multiple permeable ions. RT = ln P [ ] [ ] + + Na o + PK K [ Na ] [ ] + i + P K i Na E rest + F P Na K Goldman-Hodgkin Katz Voltage Equation o

42 The mechanism of action potential generation Historical Background Hodgkin and Huxley Following Hodgkin and Katz (1949), the big remaining question was how is the permeability of the membrane to specific ions linked to time and V m? This was not answered until the tour-de-force of physiology and modeling presented in four papers in 1952 by Hodgkin and Huxley. This work represents one of the highest-points in cellular biophysics and the quantitative model they developed forms the basis for understanding and modeling the excitable behavior of all neurons. Hodgkin and Huxley realized that by manipulating the ionic concentrations, combined with the techniques of the space and voltage clamps, they could disentangle the temporal contributions of different ions assuming that they responded differently to changes in V m. Removing Na + from the bathing medium, I Na becomes negligible so I K can be measured directly. Subtracting this current from the total current yielded I Na. Disentangling the ionic currents (reproduced from Hodgkin and Huxley, 1952a)

43 Historical Background Neher and Sakmann Ion channels Following Hodgkin & Huxley s results in the 1950 s two classes of transport mechanisms competed to explain their results: carrier molecules and pores - and there was no direct evidence for either. It was not until the 1970 s that the nicotinic ACh receptor and the Na + channel were chemically isolated, purified, and identified as proteins. The technical breakthrough of the patch-clamp techniques developed by Neher and Sakmann (1976) allowed them to report the first direct measurement of electrical current flowing through a single channel for which they received the 1991 Nobel prize. Patch-clamp recording from a single ACh-activated channel on a cultured muscle cell with the patch clamped to -80mV. Openings of the channel (downward events) caused a unitary 3 na current to flow, often interrupted by a brief closing. Notice the random openings and closing, characteristic of all ion channels. Fluctuations in the baseline are due to thermal noise. Reproduced from Sigworth FJ (1983) An example of analysis in Single Channel Recording, eds. Sakmann B, Neher E. Pp Plenum Press.

44 The Hodgkin-Huxley Formalism Basic Assumptions V m Im Iionic C m R m g Na g K E m E Na E K I ( t) = I ( t) m ionic + C m dv dt m I = I + I + ionic Na K I leak

45 The Hodgkin-Huxley Formalism Ohmic Currents V m C m I Na R m g Na g K E m E Na E K Ohm s law Currents are linearly related to the driving potential V m I Na ( t) = g ( V ( t), t) ( V ( t) E ) Na Na The Nernst Equation The Nernst potential, here for Na +, gives the reversal potential E Na or the ionic battery it is a function of the intra- and extracellular concentrations of the ion E Na = RT zf ln [ Na] o [ Na] i

46 The Hodgkin-Huxley Formalism Voltage-Dependence of Conductances Experimentally recorded (circles) and theoretically calculated (smooth curves) changes in g Na and g K in the squid giant axon at 6.3C C during depolarizing voltage steps away from the resting potential (here set to 0). Inactivation is demonstrated by the decay of g Na following its initial rise. Reproduced from Hodgkin AL (1958) Ionic movements and electrical activity in giant nerve fibres, Proc R Soc Lond B 148:1-37

47 The Hodgkin-Huxley Formalism Gating Particles V m C m I Na R m g Na g K E m E Na E K g g Na K = = g g K Na n 3 m h Gating particles (m,h,n, etc.) were introduced to describe the dynamics of the conductances (time- and voltage-dependent) and scale a maximal conductance. They can be activating or inactivating. The values range from 0 to 1 and (knowing what we know today with respect to ion channels) can be thought of as the percentage of channels in the activated or inactivated state. 4

48 The Hodgkin-Huxley Formalism Gating particles obey first order kinetics p i = probability (or fraction of) gate(s) i being in permissive state (1-p i ) = probability (or fraction of) gate(s) i being in non-permissive state i i i i i p V p V dt dp ) ( ) )(1 ( β α = Steady state solution ) ( ) ( ) ( ) (, V V V V p i i i t i β α α + = ) ( ) ( 1 ) ( V V V i i i β α τ + = Time constant

49 Activation and Inactivation Kinetics Potassium Current I K Non-inactivating current I K = g K n 4 ( V E ) K Activation particle n dn dt = αn ( V )(1 n ) βn( V ) n i.e. dn dt n = τ n n Time-dependent solution n t τ n ( t) n ( n n ) e = 0 Hodgkin and Huxley s Parameterization g K α β n = 36 ms/cm ( V ) V = V ( (10 )/10 e 1) V / 80 ( V ) = e

50 Activation and Inactivation Kinetics Sodium Current I Na Activating and inactivating current I Na = g Na 3 m h ( V E ) Na Gating particles m and h dm dt m τ m dh = = m dt τ h h h activation inactivation Hodgkin and Huxley s Parameterization α β m m ( V ) 10 g Na 25 V = V V / 18 ( V ) = 4 e ( (25 )/10 e 1) =120 ms/cm α β h h 2 V / 20 ( V ) = 0.07 e ( V ) 1 = (30 V )/10 e + 1

51 Activation and Inactivation Kinetics Graphical Representation τ τ τ m h n Time constants (upper plot) and steady-state activation and inactivation (lower plot) as a function of the relative membrane potential V for sodium activation m (solid line) and inactivation h (long dashed line) and potassium activation n (short dashed line). m h n m Reproduced from Koch C (1999) Biophysics of Computation, Oxford University Press.

52 Generation of Action Potentials The Complete Hodgkin-Huxley Model Computed action potential in response to a 0.5 ms current pulse of 0.4 na amplitude (solid lines) compared to a subthreshold response following a 0.35 na current pulse (dashed lines). (A) Time course of the two ionic currents note their large size relative to the stimulating current (B) Membrane potential in response to threshold and subthreshold stimuli (C) Dynamics of the gating particles note that the Na + activation m changes much faster than h or n Reproduced from Koch C (1999) Biophysics of Computation, Oxford University Press.

53 Generation of Action Potentials The Complete Hodgkin-Huxley Model Results of the complete model: 1) Action potential generation 2) Threshold for spike initiation 3) Refractory period For an overview on the Loligo s axon (Giant squid acon) see

54 Activation and Inactivation Kinetics Temperature Dependence Q 10 Kinetics of channels/currents (i.e. α and β) are strongly dependent on temperature while the peak conductance remains unchanged be very careful when reading the methods section of a neurophysiology paper!!! Hodgkin and Huxley recorded from the Loligo axon at 6.3 C and so the rate constants shown above are for that temperature To adjust for different temperature, α and β must be scaled by ( T Q 10 T measured )/10 Where the Q 10 measures the increase in the rate constant for every 10 C change from the temperature at which the kinetics were measured this is typically between 2 and 4

55 The Hodgkin-Huxley Formalism Summary 1) The Hodgkin-Huxley 1952 model of action potential generation and propagation is the single most successful quantitative model in neuroscience 2) The model represents the cornerstone of quantitative models of neuronal excitability 3) The heart of the model is a description of the time- and voltage-dependent conductances for Na + and K + in terms of their gating particles (m, h, and n) 4) Gating particles can be of the activation or inactivation variety activation implies its amplitude (from 0 to 1) increases with depolarization while the converse is true of inactivation 5) Kinetics of gating are represented either by the rate constants α and β or the steady-state activation/inactivation and time constant (e.g. n and τ n ) 6) Without any a priori assumptions about action potentials, this model generates APs of appropriate shape, threshold and refractory periods (both absolute and relative) 7) Temperature can have a dramatic effect on the kinetics of gating and, ideally, should be accounted for in a model by incorporation of the Q 10 scaling factor this is an experimentally-determined quantity

56 JHU BME Physiological Foundations Lecture 8 (Neural Computation 1) Kechen Zhang 02/23/2005

57 Commonly Used single Neuron Models Compartment models Integrate-and-fire models Stochastic models Firing rate models

58 Compartment model of various complexity Details of 3 compartments

59 Integrate-and-Fire Model dv ( V V C = ) 0 + dt R I Generate a spike whenever V reaches threshold V thres, then immediately reset it to V reset V - voltage I - input current C - capacitance R -resistance V 0 - resting potential

60 Input-output relation (gain function) of an integrate-and-fire neuron

61 Firing Rate Model of a Neuron Inputs firing rates from other neurons: x 1, x 2, x 3, Synaptic weights: w 1, w 2, w 3, Nonlinear gain function (input-output relation): g Threshold parameter: q Output firing rate: y = g w i xi θ i

62 Feedforward vs. Recurrent Networks Feedforward Recurrent

63 An Example of Biological Neural Networks: C elegans All synaptic connections among 302 neurons are known and consistent from animal to animal A feedforward network can describe only part of a larger recurrent network

64 An Example of Biological Neural Networks: Mammalian Neocortex Neocortex of a kitten (Cajal) Local architecture Over 80% of brain volume in humans are occupied by neocortex, and over 98% of axons in white matter interconnect different areas of the neocortex itself rather than connect with other parts of the brain. So the neocortical system is a huge recurrent network.

65 Example: A feedforward network in visual pathway Oriented receptive field of a simple cell in visual cortex can be derived by linearly combining inputs from many smaller circular receptive fields in lateral geniculate body. (Hubel)

66 McCulloch and Pitts Model (1943) Binary neurons with thresholds Synchronized update Memory as reverberant activity in a circle A network is powerful enough to do arbitrary logical calculations But, brain is not a digital computer

67 Perceptron Invented by Rosenblatt (1950 s) Output is a weighted linear combination of the inputs Supervised learning: minimize error for each example by updating the weights Guaranteed convergence to a solution in finite steps if the classification problem is linearly separable

68 Output: Perceptron y = where i w x i i θ input pattern: x 1, x 2, x 3, weights: w 1, w 2, w 3, threshold: q Each input pattern is classified into one of two classes depending on whether y > 0 or y < 0. Learning rule: w i where = η( Y y) x desired output: Y learning rate: h > 0 i

69 Example of Linear Separability: Volleyball team Name Height (input x 1 ) Weight (input x 2 ) Gender (desired output Y) John Mary Tom Ouput y Weight = w1x1 + w2x2 θ Separable Case where θ = 0.5 is threshold Weight Inseparable Case y > 0 y < 0 Height Height

70 Perceptron learning rule as gradient descent Consider the square error for the desired output Y and the actual output 1 2 E = ( y Y ) 2 To minimize the error, we change the weights slightly along the direction of descent (gradient descent ). w i E = η w i = η ( y Y ) y w the steepest This gives : = η ( y Y ) x which is exactly the perceptron learning rule. i y : i

71 Exclusive-OR (XOR) Problem Input x 1 Input x 2 Output y x x 1 XOR: either A or B but not both simple example of a linearly inseparable problem cannot be learned by a perceptron because it is linear.

72 A Solution to the XOR Problem by Multilayer Perceptron Multiple solutions can be found by training a network. Here is one solution. A number inside a circle is a threshold and a number by an arrow is a synaptic weight.

73 Multilayer Perceptron Output Units Hidden Units Input Units

74 Multilayer Perceptron Learning A multilayer feedforward network learns to approximate an unknown input-output relationship from given examples of inputoutput pairs. All the weights in the network can be learned by minimizing the square error between the actual outputs of the network and the desired outputs in the examples, just like in a single layer perceptron. There are several algorithms for minimizing the error. The first one, the back-propagation algorithm, was discovered by Rumelhart, Hinton and McCleveland in mid 1980 s. It is essentially a gradient descent method, like the learning rule of the single layer perceptron. Unlike in a single layer perceptron, the final form of the learning rule here is not local in the sense that modification of a synaptic weight depends not only on the activities of the pre- and postsynaptic neurons but also on the activities of all other neurons in the entire network.