10-07-15. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 1 Development of Complex Curricula for Molecular Bionics and Infobionics Programs within a consortial* framework** Consortium leader PETER PAZMANY CATHOLIC UNIVERSITY Consortium members SEMMELWEIS UNIVERSITY, DIALOG CAMPUS PUBLISHER The Project has been realised with the support of the European Union and has been co-financed by the European Social Fund *** **Molekuláris bionika és Infobionika Szakok tananyagának komplex fejlesztése konzorciumi keretben ***A projekt az Európai Unió támogatásával, az Európai Szociális Alap társfinanszírozásával valósul meg. PETER PAZMANY CATHOLIC UNIVERSITY SEMMELWEIS UNIVERSITY 10-07-15. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 1 2011.10.15. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 2 Peter Pazmany Catholic University Faculty of Information Technology BEVEZETÉS A FUNKCIONÁLIS NEUROBIOLÓGIÁBA INTRODUCTION TO FUNCTIONAL NEUROBIOLOGY www.itk.ppke.hu By Imre Kalló Contributed by: Tamás Freund, Zsolt Liposits, Zoltán Nusser, László Acsády, Szabolcs Káli, József Haller, Zsófia Maglóczky, Nórbert Hájos, Emilia Madarász, György Karmos, Miklós Palkovits, Anita Kamondi, Lóránd Erőss, Róbert Gábriel, Kisvárdai Zoltán Introduction to functional neurobiology: Neuronal modelling 2011.10.15. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 3 www.itk.ppke.hu Neuronal modelling Szabolcs Káli Pázmány Péter Catholic University, Faculty of Information Technology Infobionic and Neurobiological Plasticity Research Group, Hungarian Academy of Sciences –Pázmány Péter Catholic University –Semmelweis University 10-07-15. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 4 Neural modeling Types of models: • Descriptive –What is it like? • Mechanistic –How does it function? • Explanatory –Why is it like that? 10-07-15. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 5 Howisinformationencodedbyactionpotentialtrains? Neuronsrespondtoinputtypicallybyproducingcomplexspikesequencesthatreflectboththeintrinsicdynamicsoftheneuronandthetemporalcharacteristicsofthestimulus. Simpleway:Counttheactionpotentialsfiredduringstimulus,repeatthestimulusandaveragetheresults: Some fundamental questions Picture: Recordings from the visual cortex of a monkey. A bar of light was moved through the receptive field of the cell at different angles (figure A). The highest firing rate was observed for input oriented at 0 degrees (figure B). 10-07-15. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 6 How to decode information encoded by action potential trains? Example: Arm movement position decoding: If we take the average of the preferred directions of the neurons weighted by their firing rates, we get the arm movement direction vector. Some fundamental questions Picture: Comparison of arm position and arm position-sensitive neurons. The population activity was recorded in 8 directions. Arrows indicate vector sums of preferred directions, which is approximately the arm movement direction. 10-07-15. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 7 Why does a specific part of the brain use a specific type of coding? Example: Visual input noise filtering in ganglion cells: The structure of the receptive field changes according to the input signal-to noise ratio. Some fundamental questions Solid curves are for low noise input (bright image), dashed lines are high noise input. Left: The amplitude of the predicted Fourier-transformed linear filters. Right: The linear kernel as a function of the distance from the center of the receptive field 10-07-15. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 8 How do neurons as information processing units function; specifically, what is the relation between the temporal and spatial pattern of the input and the spatial and temporal pattern of the output? Example: Some fundamental questions Picture:theeffectsofconstantsustaineddendriticcurrentinjectioninahippocampalpyramidalcell.Thecellrespondswithaburstofspikes,thensustainedspiking.Indistalregionsonlyaslow,large-amplitudeinitialresponseisvisible,correspondingtoadendriticcalciumspike. 10-07-15. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 9 How do neurons communicate, and what collective behaviors emerge in networks? Example: Orientation selectivity and contrast invariance in the primary visual cortex Some fundamental questions Left picture: schematics of a recurrent network with feedforward inputs. Middle picture: The effect of contrast on orientation tuning. Figure A: orientation-tuned feedforward input curves for 80%,40%,20%,10% contrast ratios. Right picture: The output firing rates for response to input in figure A. Due to network amplification, the response of the network is much more strongly tuned to orientation as a result of selective amplification by the recurrent network, and tuning width is insensitive to contrast. 10-07-15. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 10 How does cellular-level (synaptic) plasticity function? How can we understand behavioral-level learning? What is the connection between the two? Example: The development of ocular stripes in the primary visual cortex Left picture: schematics of the model network where right-and left-eye inputs from a single retinal location drive an array of cortical neurons. Right picture: Ocular dominance maps, the light and dark areas along the cortical regions at the top and bottom indicate alternating right-and left-eye innervation. Top: In vitro measurements. Bottom: The pattern of innervation for the model after Hebbian development. Some fundamental questions 10-07-15. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 11 V: Membrane potential [V]. Ie: Current injected into the cell, with an electrode for example [A]. Q: Excess internal charge [C]. Rm: Membrane resistance, treated as a constant in the equations (specific membrane resistance, rm) [Ohm]. Cm: Membrane capacitance, treated as a constant in the equations (specific membrane capacitance, cm) [F]. The cell membrane is represented by a resistance and a battery in parallel with a capacitance. 10-07-15. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 12 ThetimederivativeofthechargedQ/dtisequaltothecurrentpassingintothecell,sotheamountofcurrentneededtochangethemembranepotentialofaneuronwithatotalcapacitanceCmataratedV/dtisCmdV/dt. Thisisequalto: WhereEristherestingpotentialofthecell.Inmostequationsmembraneconductance(gm)isusedinsteadofresistance(gm=1/rm),becauseitiscorrelatedwithbiophysicalpropertiesoftheneuron: 10-07-15. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 13 The product of the membrane capacitance and the membrane resistance is a quantity inunits of time,called the membrane time constant, denoted by tau: The membrane time constant sets the basic time scale for changes in the membrane potential and typically falls in the range between 10 and 100 milliseconds. The total membrane conductance can change dynamically, causing the membrane time constant to change, too. 10-07-15. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 14 Example: 10-07-15. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 15 Where gi(V-Ei)is the current flowing through ion channel i. Voltage-dependent conductances Singleionchannelsareeitherinanopenoraclosedstate.Theprobabilityofthestatescandependonthemembranepotentialandonthebindingofvarioussubstances(e.g.neurotransmitters)tothecellmembrane. 10/15/2011 TÁMOP –4.1.2-08/2/A/KMR-2009-0006 16 Figure:Thecurrentspassingthroughasingleionchannel.Intheopenstate,thechannelpasses-6.6pAattheholdingpotentialof-140mV.Thisisequivalenttomorethan107chargespersecondpassingthroughthechannel. 10/15/2011 TÁMOP –4.1.2-08/2/A/KMR-2009-0006 17 ThismeansfortheK+ionstopass,4gatesneedtobeopen. Example: In the case of the "delayed rectifier" K+current: Conductance per unit area of membrane for channel type i: Where Piis the probability of a single channel being in the open state. This is approximately equal to the fraction of open channels (if the number of channels is large). is the maximal conductance per unit area of membrane. Structurally, ion channel pores have several gates, which all need to be open for current to flow through the channel. The Hodgkin-Huxley model Gating equation 10/15/2011 TÁMOP –4.1.2-08/2/A/KMR-2009-0006 18 The transition of each gate is described by a kinetic scheme in which the gating transition closed-to-open occurs at voltage-dependent rate and the reverse transition occurs at rate nis the probability that we find a gate open. Left: example transition functions plotted as a function of the membrane potential 10/15/2011 TÁMOP –4.1.2-08/2/A/KMR-2009-0006 19 Gating equation (other form) The same equation in a useful form: where is called the time constant and is the steady-state activation function. Phenomenological models of synaptic conductances • Exponential (for example AMPA type glutamate receptor) • Difference of exponentials (for example GABAA). This method uses two time constants, thus both rise and decay can be described • Alpha function 10/15/2011 TÁMOP –4.1.2-08/2/A/KMR-2009-0006 20 This equation describes an isolated presynaptic release that occurs at t=0, reaches its maximum at the time constant, and decays with the time constant. Figure: Example alpha function with Pmax=200 and tau=30. 10/15/2011 TÁMOP –4.1.2-08/2/A/KMR-2009-0006 21 The NMDA-type glutamate synaptic receptor FigureA:Current-voltagerelationsoftheNMDAchannelintheabsenceandinthepresenceofmagnesium. FigureB: Jaggedline:ExperimentallyrecordedNMDAcurrent. Dottedline:computedNMDAcurrent,withadoubleexponentialtimecourse. TheNMDAconductancedependsnotonlyonthebindingofglutamate,butalsoonpostsynapticvoltage(andtheextracellularMg2+concentration). The NMDA-type glutamate synaptic receptor TodescribetheMg2+dependenceoftheNMDAchannelanadditionalfactorisintroduced,whichdependsonthepostsynapticpotential: 10/15/2011 TÁMOP –4.1.2-08/2/A/KMR-2009-0006 22 WhereP(t)istheopenprobabilityfactor,andGNMDA(V)describesanextravoltagedependenceduetothefactthatneartherestingpotentialNMDAreceptorsareblockedbyMg2+ions,sotoactivatetheconductancethepostsynapticneuronneedstobedepolarized.Thisdependencecanbeapproximatedbyasigmoidfunction: (typical fitted parameters: 1/. = 3.57 mM, 1/. = 16.13 mV) 10/15/2011 TÁMOP –4.1.2-08/2/A/KMR-2009-0006 23 TheNMDAchannelalsoconductsCa2+ions,whichisimportantinplasticity. The NMDA-type glutamate synaptic receptor Figure:DependenceoftheNMDAconductanceonthemembranepotentialatdifferentMg2+concentrations. 10-07-15. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 24 •Lots of different types of ion channels • Complicated and cell-type dependent morphology • The temporal and spatial pattern of synaptic input • Neuromodulators, intracellular messenger molecules,... Causes ofmore complex behavior in real neurons Biophysically detailed multicompartmental models Figure: The electrical circuit representation of three compartments in a multicompartmental model. Circles with V represent voltage-gated conductances. 10-07-15. Biophysically detailed multicompartmental models The current flowing through compartment where is the total injected electrode current. is the total surface area of the compartment. are the resistive couplings to the neighboring compartments. , with neighbors and is the membrane potential of the compartment. is the transmembrane current. is the membrane capacitance of the compartment. 10-07-15. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 27 If the membrane potential reaches a threshold, the cell fires an action potential, and the membrane potential is set toa „reset potential”. Simplest firing model: passive „integrate-and-fire” Integrate-and-fire model Or: Picture: Membrane potential trace of an integrate-and-fire model (Ie is the injected current). 10-07-15. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 28 In the case of constant current injection, the analytic solution: Integrate-and-fire model Picture: integrate-and fire models compared to in vitro recordings. Left: Firing rate as a function of the injected current. Continuous line: model results; filled circles: Results for the first two spikes fired, in vivo recordings; open circles: steady-state firing frequency, in vivo recordings. Middle: In vivo recording from pyramidal cell. Right: voltage trace of the adaptive integrate-and-fire model. The adaptation of the firing rate and refractory states are relatively easy to implement. Firing-rate-based models 10/15/2011 TÁMOP –4.1.2-08/2/A/KMR-2009-0006 29 Picture: Synaptic inputs to a single neuron. u: Input rate vector. Denotes firing rate, not the membrane potential. w: Synaptic (input) weight vector. For excitatory synapses w(i)>0, for inhibitory w(i)<0 v: Output rate vector. In the case of a single neuron, it has only one element. 10/15/2011 TÁMOP –4.1.2-08/2/A/KMR-2009-0006 30 Ks(t) is the synaptic kernel function, which describes the time course of the synaptic current in response to a presynaptic spike arriving at time t=0. Properties: Synaptic kernel Example: If an action potential arrives at t=0at input b, the synaptic current generated at the postsynaptic neuron at time tis wbKs(t), where Ks(t)is the synaptic kernel. 10/15/2011 TÁMOP –4.1.2-08/2/A/KMR-2009-0006 31 1. Calculating the total (somatically measured) synaptic current where is the neural response function, which describes the sequence of spikes fired by presynaptic neuron b. is the Dirac-delta function. Nuis the number of input neurons. tb,iis the time when a presynaptic spike occurs at input b. The total synaptic current at time t: 10/15/2011 TÁMOP –4.1.2-08/2/A/KMR-2009-0006 32 If the synaptic kernel is exponential: Then differentiating (1) gives: If the neuron response function is replaced by the firing rate of neuron b, denoted by (1) 1. Calculating the total (somatically measured) synaptic current 2. Calculating the firing rate: activation function To complete the firing rate model we must determine the postsynaptic firing rate vfrom Is. For a constant input v=F(IS), where Fis the activation function. It can be a sigmoid function, or -most frequently-linear function with a threshold: 10/15/2011 TÁMOP –4.1.2-08/2/A/KMR-2009-0006 33 where is the threshold (measured in Hz), and is the half wave rectification operator: For any x For convenience we assume that Isis multiplied by a constant which converts nA to Hz. 10/15/2011 TÁMOP –4.1.2-08/2/A/KMR-2009-0006 34 Steady-state output fire rate If all the inputs are time-independent (or in steady-state), the total current is: Is=wu. As a consequence the steady-state output firing rate is: Where wis the synaptic weight vector and uis the synaptic input vector. This equation describes how the neuron responds to constant (time-independent) current. 10/15/2011 TÁMOP –4.1.2-08/2/A/KMR-2009-0006 35 If IS(t)is time-dependent: and In this case it is assumed that the firing rates follow the time-varying currents instantaneously. Other model: The time-dependent firing rate is modeled as a low-pass filtered version of the steady-state firing rate: Firing-rate model with time-dependent dynamics Where is a time constant that determines how rapidly the firing rate approaches its steady-state value for constant Is. 10/15/2011 TÁMOP –4.1.2-08/2/A/KMR-2009-0006 36 If meaning that the firing rate time constant is much largerthan the synaptic rate time constant, then we can replace the time-dependent synaptic current function ( Is(t) ) with the total steady-state current (wu): In other words, it is assumed that the firing rate is a low-pass filtered version of the input current. Firing-rate model with time-dependent dynamics Feedforward and recurrent networks 10/15/2011 TÁMOP –4.1.2-08/2/A/KMR-2009-0006 37 Picture A: Feedforward network. Picture B: Recurrent network with feedforward inputs. W: Feedforward synaptic weight matrix, where Wabis the strength of the synapse from input unit bto output unit a. M: Synaptic weight matrix for the recurrent layer. Feedforward and recurrent networks Output firing rates in feedforward networks (fig. A): or Adding the recurrent connections (fig. B): 10/15/2011 TÁMOP –4.1.2-08/2/A/KMR-2009-0006 38 If we distinguish excitatory and inhibitory populations: (h=Wu) Another frequent abstraction: (symmetricrecurrentconnections)—>simplerdynamics Thesetwoassumptionsareseeminglycontradictory,butcanbereconciledifweassumethatinhibitionismuchfasterthanexcitation.Thensteady-stateinhibitoryactivitycanbesubstitutedintothefirstequation,andtheeffectiveinteractionbetweenexcitatoryneuronscanbesymmetricwithappropriateweightmatrices. 10/15/2011 TÁMOP –4.1.2-08/2/A/KMR-2009-0006 39 Coordinate transformation in feedforward networks Reaching for a visible object requires a transformation from (retinal) sensory coordinates to body-centered motor coordinates. 10/15/2011 TÁMOP –4.1.2-08/2/A/KMR-2009-0006 40 sis the location of the target in retinal coordinates. gis the gaze angle, indicating the direction of gaze relative from the axis of the body. s+gis the direction of the target relative to the body. 10/15/2011 TÁMOP –4.1.2-08/2/A/KMR-2009-0006 41 In the premotor area of the frontal lobes Neural coordinate transformations Picture: Firing rates of a visually responsive neuron in the premotor cortex of a monkey. Visual stimuli were incoming objects at various angles. A: The response tuning curve does not change with eye position. B: If the monkey’s head is turned 15 degrees to the right, the tuning curves shift by 15 degrees too. C: Model turning curves at -20,0,10 degrees. (dotted, solid, heavy dotted) Possibleintermediaterepresentation(inarea7aoftheparietallobe) Question:Canwecombinetheactivityofsuchneuronsinafeedforwardnetworktocreateoutputneuronsfiringinbody-centeredcoordinates? CanbemodeledbytheproductofaGaussianfunctionofsandasigmoidfunctionofg;theactivityofvariousneuronsofthepopulationis is the center of the sigmoid. is the mean of the Gaussian and where 10/15/2011 TÁMOP –4.1.2-08/2/A/KMR-2009-0006 42 Linear recurrent networks Multiplying by the eigenvector Let us assume that Mis symmetric —> real eigenvalues and orthogonal eigenvectors forming a basis, so we may write and substituting into (1), 10/15/2011 TÁMOP –4.1.2-08/2/A/KMR-2009-0006 43 (1) This transformation could also be implemented by a purely feedforward network! For time-independent inputs the solution is: If for any ., the network is unstable. If exponentially approaches the stationary value with the time constant 10/15/2011 TÁMOP –4.1.2-08/2/A/KMR-2009-0006 44 Linear recurrent networks The steady-state value for v(t) is: Applications of linear recurrent networks 1. Selective amplification Assume that is very close to 1, and all other eigenvalues are signif- icantly smaller. Then the steady state is 10/15/2011 TÁMOP –4.1.2-08/2/A/KMR-2009-0006 45 The response of the network is dominated by the projection of the input vector along the axis defined by e1, in other words it encodes an amplified version of the input onto e1. Strength of amplification: Continuously parametrized neuronal populations Picture:Inprimaryvisualcortex,aneuronmaybecharacterizedbytheorientationofitspreferredstimulus(.).Theseneuronstendtofireatthehighestfrequenciesifthestimulusisataspecifiedangleintheirreceptivefield. Tomodelsuchnetworks,itismoreefficienttoindexneuronsbytheir(continuouslyvarying)preferredparameters. Wecanreplacethediscretepopulationwithacontinuouspopulation: firing rates, synaptic weights. Then the equation describing the dynamics becomes: 10/27/2010 TÁMOP –4.1.2-08/2/A/KMR-2009-0006 46 10-07-15. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 47 Applications of linear recurrent networks: Selective If the neurons are described by a preferred value of a periodic variable (.), and M(.,.’)~cos(.-.’), the network selectively amplifies the first Fourier component of the input: Picture A: The input as a function of the preferred angle B: The activity of the network as a function of the preferred angle. Input stimulus was the same as in picture A. C: The Fourier transform amplitudes of the input. D: The Fourier transform amplitudes of the output. 10/15/2011 TÁMOP –4.1.2-08/2/A/KMR-2009-0006 48 Forh=0, the network sustains its activity, which acts as a memory of the integral of previous inputs. In other words the network "remembers" the previous state. 2. Input integration (h=Wu) Example:networkforstoringeyeposition,whichintegratestheoutputofbrainstemocularmotorneurons Picture:Integratorneuronactivitythatisinvolvedinhorizontaleyepositioning. Problem:theeigenvaluemustbereallycloseto1 10/15/2011 TÁMOP –4.1.2-08/2/A/KMR-2009-0006 49 2. Input integration Nonlinear recurrent networks In biological neural networks firing rates must be positive. Taking this into account: where The previous continuous model now becomes 10/15/2011 TÁMOP –4.1.2-08/2/A/KMR-2009-0006 50 is the half wave rectification operator. Selective amplification in the nonlinear case: 10/15/2011 TÁMOP –4.1.2-08/2/A/KMR-2009-0006 51 Picture: Selective amplification in a linear network with A: The noisy input as a function of the preferred angle B: The steady-state output as a function of the preferred angle. Input stimulus was the same as in picture A. C: The Fourier transform amplitudes of the input. D: The Fourier transform amplitudes of the output. 10/15/2011 TÁMOP –4.1.2-08/2/A/KMR-2009-0006 52 Orientation selectivity and contrast invariance in V1 Selective amplification in the nonlinear case: Recurrent model of simple cells in the primary visual cortex. Picture: The effect of contrast on orientation tuning A: The feedforward input as a function of preferred orientation. Contrast ratios are (from top to bottom): 80%, 40%, 20%, 10% B: The output firing rates in response to the inputs in figure A. C: Tuning curves measured experimentally. Input selection in nonlinear networks: Winner-takes-all input selection. 10/15/2011 TÁMOP –4.1.2-08/2/A/KMR-2009-0006 53 Figure A: The input to the network consisting of two peaks. Figure B: Network response. The output has a single peak at the location of the higher of the two peaks of the input. 10/15/2011 TÁMOP –4.1.2-08/2/A/KMR-2009-0006 54 (as seen earlier in area 7a of the parietal lobe): Input selection in nonlinear networks: Gain modulation Picture: Effect of adding a constant to the input. Figure A: The input to the network with one peak, with different amounts of added gain input. Figure B: Network response. The higher gain yields a higher output. Application:short-term(working)memory;thenetwork"remembers"theprecedingstimulusevenintheabsenceofexternalinput. 10/15/2011 TÁMOP –4.1.2-08/2/A/KMR-2009-0006 55 Input selection in nonlinear networks Picture: response to varying input. A,B: Input and output, with constant background and localized excitation. C,D: After switching to a constant input, the response characteristics are the same Maximum likelihood estimation Theabovecomputationscanalsobeinterpretedas(nonlinear)regression,andmaybeusedtoapproximatethe"maximumlikelihood"estimateofthevalueencodedbyanoisyinput. Thestandarddeviationoftheestimateis • 4.5ofor a simple "vector decoder" • 1.7ofor recurrent network "cleanup" followed by vector decoding •0.88ofor the optimal (real maximum likelihood) decoder (Cramer-Raolower bound) 10/15/2011 TÁMOP –4.1.2-08/2/A/KMR-2009-0006 56 Associative (long-term) memory Aswehaveseen,recurrentnetworksoftenhaveactivitypatternswhichbehaveasfixedpointattractors.Thesemayalsobeconsideredasstoredmemorytraces,providedthatwecanspecifythefixedpoints(preferablyviaaplausibleactivity-basedlearningrule). Autoassociativefunction:thedynamicsofthenetworkreconstructstheoriginalpatternbasedonafragmentoranoisyversion. 10/15/2011 TÁMOP –4.1.2-08/2/A/KMR-2009-0006 57