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Networks of neurons • Neurons are organized in large networks. A typical neuron is cortex receives thousands of inputs. • Aim of modeling networks: explore the computational potential of such connectivity. Models of Networks of Neurons - What computations? (e.g. gain modulation, integration, selective amplification of some signal, memory etc..) - What dynamics ? (e.g. spontaneous acticity, variability, oscillations) •Tools: - models of neurons and synapses : spiking neurons (IAF) or firing rate - analytical solutions, numerical integration 1.1 Introduction 3 B A dendrite What’s a network ? apical dendrite ~80% excitatory cells (pyramidal neurons), • In cortex, 1.1 Introduction 3 ~20% inhibitory soma neurons (smooth stellate + large variety of other types)/ a.k.a interneurons. soma A B axon C basal dendrite dendrite apical dendrite dendrite axon soma axon collaterals soma axon basal dendrite axon soma C dendrite axon Figure 1.1:axon Diagrams of three neurons. A) A cortical pyramidal cell. These are collaterals the primary excitatory neurons of the cerebral cortex. Pyramidal cell axons branch soma locally, sending axon collaterals to synapse with nearby neurons, and also project more distally to conduct signals to other parts of the brain and nervous system. axon B) A Purkinje cell of the cerebellum. Purkinje cell axons transmit the output of the cerebellar cortex. C) A stellate cell of the cerebral cortex. Stellate cells are one of a large class of cells that provide inhibitory input to the neurons of the Figure 1.1: Diagrams of three neurons. A) A cortical pyramidal cell. These are cerebral cortex. To give an idea of scale, these figures are magnified about 150 fold. the primary excitatory neurons of the cerebral cortex. Pyramidal cell axons branch (Drawings from Cajal, 1911; figure fromwith Dowling, 1992.) and also project locally, sending axon collaterals to synapse nearby neurons, more distally to conduct signals to other parts of the brain and nervous system. B) A Purkinje cell of the cerebellum. Purkinje cell axons transmit the output of the cerebellar cortex. C) A stellate cell of the cerebral cortex. Stellate cells are one of a large class of cells that provide inhibitory input to the neurons of the What’s a network ? • Laminar Organization. Cortex is divided into 6 layers. Models usually pool all layers together. What’s a network ? Connectivity • 3 types of connections: feed-forward, recurrent (lateral), feedback. • Columnar Organization. se on esp nR tio Neurons in small (30-100 micrometers) columns perpendicular to the layers ula op B! P (across all layers) respond to similar stimulus features. 50 ves A! g nin Tu 180 r Cu 90 25 0 !90 0 ! test V4 0 !18 50 s 180 90 25 Re nse spo IT 0 180 ! !90 V2 0 ! test V1 Retina 6 FEEDBACK • Excitatory • Modulatory influence V1large • Convey information from area of visual field • V2, MT • role unclear V1 LGN • Excitatory • precise topographically • debated how strong 7 LGN 8 Local Projections Long-range Horizontal Projections V1 V1 • Excitatory + Inhibitory : lots !! • < 1 mm • connectivity depend on distance, not preference. • Excitatory - Modulatory • 6-8 mm (cat, monkey) - Non overlapping RF • specific/ preferences LGN V1: Neurons. !"#$ %&$ '()#*$ +,-.(/-0$ &112$ #3+/.(.,*)$ -#4*,-0$ (-5$ 676$ /-"/8/.,*)$ -#4*,-09$ LGN 9 10 :3+/.(.,*)$-#4*,-0$(*#$;,5#'#5$(0$*#<4'(*$0=/>/-<$+,-54+.(-+#?8(0#5$/-.#<*(.#?(-5?@/*#$-#4*,-0A$ B"/'#$ /-"/8/.,*)$ -#4*,-0$ (*#$ ;,5#'#5$ (0$ +,-54+.(-+#?8(0#5$ @(0.?0=/>/-<$ -#4*,-09$ !"#$ -#4*,-$ ;,5#'$(-5$=(*(;#.#*0$B#*#$.(>#-$@*,;$C,;#*0$#.$('9$D&EE7F9$:(+"$+,*./+('$-#4*,-$/0$;,5#'#5$(0$($ V1: Neurons. !"#$ %&$ '()#*$ +,-.(/-0$ &112$ #3+/.(.,*)$ -#4*,-0$ (-5$ 676$ /-"/8/.,*)$ -#4*,-09$ 0/-<'#$G,'.(<#$+,;=(*.;#-.$/-$B"/+"$."#$;#;8*(-#$=,.#-./('$/0$</G#-$8)$ :3+/.(.,*)$-#4*,-0$(*#$;,5#'#5$(0$*#<4'(*$0=/>/-<$+,-54+.(-+#?8(0#5$/-.#<*(.#?(-5?@/*#$-#4*,-0A$ $ B"/'#$ /-"/8/.,*)$ -#4*,-0$ (*#$ ;,5#'#5$ (0$ +,-54+.(-+#?8(0#5$ @(0.?0=/>/-<$ -#4*,-09$ !"#$ -#4*,-$ $%! D& F $ %& ( !#strategies ! & % # !# " !%! D& F % )(1) Network'"modeling :HIJ! " % & ( !# ! & % # !# " !%! D& F % )JKLJM " $ $ D&6F$ $& # # ;,5#'$(-5$=(*(;#.#*0$B#*#$.(>#-$@*,;$C,;#*0$#.$('9$D&EE7F9$:(+"$+,*./+('$-#4*,-$/0$;,5#'#5$(0$($ % ( N:OP !%! D& F % )N:OP " % ( OLQ ! & "!%! D& F % )OLQ " 9 0/-<'#$G,'.(<#$+,;=(*.;#-.$/-$B"/+"$."#$;#;8*(-#$=,.#-./('$/0$</G#-$8)$ $ IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 14, NO. 6, NOVEMBER 2003 Network modeling strategies (2) we choose the ratio of excitatory to inhibitory ne we make inhibitory synaptic connections stronge input, each neuron receives a noisy thalamic inp In principle, one can use RS cells to model and FS cells to model all inhibitory neurons. Th heterogeneity (so that different neurons have d to assign each excitatory cell spike raster , where is a random v tributed on the interval [0,1], and is the neuron corresponds to regular spiking (RS) cell, and the chattering (CH) cell. We use to bias the d cells. Similarly, each inhibitory cell has and . The model belongs to the class of pulse-cou (PCNN): The synaptic connection weights bet given by the matrix , so that firing stantaneously changes variable by . e.g. integrate and fire neurons • method 1: spiking neurons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• up to 10,000 neurons. </G#-$8)$(-$('="(?@4-+./,-A$ 0=/>#$."*#0",'5$D?77$;%FA$($0=/>#$/0$#;/..#5A$."#$0=/>#$."*#0",'5$/0$#'#G(.#5$;/;/+>/-<$($*#'(./G#$ U neuron trace *#@*(+.,*)$=#*/,5$D0##$C,;#*0$#.$('9$D&EE7F$@,*$5#.(/'0FA$(-5$($P with electrophysiology, a system where all $;#5/(.#5$(@.#*?")=#*=,'(*/V(./,-$ • advantage: comparison $ DOLQF$ +,-54+.(-+#$ B(0$ (+./G(.#59$ !"#$ OLQ$ +,-54+.(-+#A$ (OLQD.FA$ ,8#)0$ ."#$ 0(;#$ #W4(./,-$ (0$ neurons can be !recorded" at all times. ( OLQ (-5$."#$04;$/0$,G#*$."#$/-5#3$!$D."#$+#''X0$,B-$0=/>#0F$*(."#*$ D&SF$#3+#=.$."(.$."#$=*#@(+.,*$/0$ ( & % & # ) long '( + ) simulations, • difficulties: lots of parameters/assumptions, $ $ $ $ D&SF$ (!# ! & " $ (!# & *.& % & *# +/ , -- #3= ,, % -- 9 $ , analysis difficult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ig. 3. Simulation of a network of 1000 randomly coupled spiking neurons. Top: spike raster shows episodes of alpha and gamma band rhythms (vertical lines). Bottom: typical spiking activity of an excitatory neuron. All spikes were first to 30 mV and then to . equalized at 30 mV by resetting % Created by Eugene M. % Excitatory neurons Ne=800; re=rand(Ne,1); [Izhikevitch, 2003] a=[0.02*ones(Ne,1); b=[0.2*ones(Ne,1); c=[-65+15*re.^2; d=[8-6*re.^2; S=[0.5*rand(Ne+Ni,Ne), Izhikevich, F Inhibitory ne Ni=200; ri=rand(Ni,1) 0.02+0.08*ri] 0.25-0.05*ri] -65*ones(Ni,1 2*ones(Ni,1)] -rand(Ne+Ni,N Network modeling strategies (3) • method 2: reduce the description to describe only rate of spiking r(t), instead of Vm(t). τr dri (t) = −ri (t) + input(t) dt Firing rate model (1) • each neuron is described at time t by a firing rate v(t). j=N ! dvi (t) wij uj ) τr = −vi (t) + F ( dt j=1 • Interpretation: average over time, average over equivalent neurons • In absence of input, the firing rate relaxes to 0 with a time constant tr - which also determines how quickly the neuron responds to input. • The input from a presynaptic neuron is proportional to its firing rate u • The weight wij determines the strength of connection of neuron j to neuron i • The total input current is the sum of the input from all external sources. Firing rate model (2) • each neuron is described at time t by a firing rate v(t). τr j=N ! dvi (t) wij uj ) = −vi (t) + F (w.u) = −vi (t) + F ( dt dot-product j=1 • F determines the steady state r as a function of input • F is called the activation function • F can be taken as a saturating function, e.g. sigmoid • F is often chosen to be threshold linear Network Architectures • A: Feedforward τr N ! dvi (t) = −vi (t) + F ( Wij uj (t)) dt j=1 • B: Recurrent τr N N ! ! dvi (t) = −vi (t) + F ( Wij uj (t) + Mik vk (t)) dt j=1 k=1 rmax F (I) = 1 + exp(g(I1/2 − I) F (I) = G[I − I0 ]+ determined by the equation τr dv = −v + F(h + M · v ) . dt (7.11) Excitatory - Inhibitory Network Neurons are typically classified as either excitatory or inhibitory, meaning that they have either excitatory or inhibitory effects on all of their postsynaptic targets. property is formalized Dale’s law, which models haveThis a single population of neuronsinand the weights are states that • Some a neuron cannot excite some of its postsynaptic targets and inhibit others. allowed to be positive and negative. In terms of the elements of M, this means that for each presynaptic neuron models represent the excitatory and inhibitory population separately. • Other a" , Maa" must have the same sign for all postsynaptic neurons a. To im(more !biological" + richer dynamics). pose this restriction, it is convenient to describe excitatory and inhibitory matrices, MEE, M • 4 weight IE, M II, MEI neurons separately. The firing-rate vectors vE and vI for the excitatory and inhibitory neurons are then described by a coupled set of equations identical in form to equation 7.11, τE dvE = −vE + FE (hE + MEE · vE + MEI · vI ) dt (7.12) dvI = −vI + FI (hI + MIE · vE + MII · vI ) . dt (7.13) Dale’s law Example: Orientation selectivity as a model problem: spiking networks and ring model excitatoryinhibitory network and τI There are now four synaptic weight matrices describing the four possible types of neuronal interactions. The elements of MEE and MIE are greater than or equal to zero, and those of MEI and MII are less than or equal to zero. These equations allow the excitatory and inhibitory neurons to have different time constants, activation functions, and feedforward inputs. In this chapter, we consider several recurrent network models described by equation 7.11 with a symmetric weight matrix, Maa" = Ma" a for all a and neurons selective to orientation, V1’s are: analysis, but symmetric coupling a" . LGN Requiring M toare be not symmetric simplifies the mathematical Origin Orientation selectivitythat ? neuron a, which is exciit violates Dale’s law.ofSuppose, for example, tatory, and neuron a" , which is inhibitory, are mutually connected. Then, Draft: December 19, 2000 Theoretical Neuroscience Model of Hubel and Wiesel (1962) Text • Hubel and Wiesel (1962) proposed that the oriented fields of V1 neurons could be generated by summing the input from appropriately selected LGN neurons. • The model accounts for selectivity in V1 on the basis of a purely feedforward architecture. Ferster and Miller — July 30, 2000 Text 4 Text V1 Sclar and Freeman, 1982 V1 • Example of a computation, emergence of a new property. LGN Figure 1: neurons V1 simple cell A. A map of the receptive field of a simple cell in the cat visual cortex. A light flashed in the ON subregion (x) or turned off in an OFF region (triangles) excites the cell, while a light flashed in an OFF region or turned off in the ON region inhibit the cell. Other arrangements of the subregions are possible, such as a central OFF region and flanking ON regions, or one ON and one OFF region. B. Hubel and Weisel’s model for how the receptive field of the simple cell can be built from excitatory input from geniculate relay cells. The simple cell (below right) receives input from output output Connectivity achieves contrast invariance through feedforward inhibition Connectivity sharpens the tuning curve input Due to the precise organization of the thalamic Due to input the imprecise afferents, to V1 is organization of the sharply tuned thalamic afferents, input to V1 is broadly tuned output input V1 Feedforward Recurrent Hubel and Wiesel, 1962; Troyer, Krukowski, Priebe and Miller, 1998 Somers, Nelson and Sur 1995; Sompolinsky and Shapley, 1997 ~ 100, 000 synapses `Feedforward’ model vs `Recurrent’ ~ 1250 conductance based IAF neurons Wee, Wie, Wei, Poisson spike trains Retina/ LGN Seriès, Latham and Pouget - Nature Neuroscience 2004 ON OFF • Explore physiological and anatomical plausibility: - cortical connectivity scheme, - thalamocortical connectivity, - properties of inhibition in Cx (inactivation) … (Sompolinsky and Shapley, 1997; Ferster and Miller, 2000). • Coding efficiency (are these models making different predictions in terms of information transmission?) 23 22 7.4 Recurrent Networks 25 1.9. This value, being Network larger thanmodels one, would lead to an unstable network - summary in the linear case. While nonlinear networks can also be unstable, the restriction to eigenvalues less than one is no longer the relevant condition. The Ring Model (1) In a nonlinear network, the Fourier analysis of the input and output remodels:astoinformative understand the of connectivity • Network sponses is no longer as implications it is for a linear network.inDue to terms of computation and dynamics. the rectification, the ν = 0, 1, and 2 Fourier components are all amplified (figure 7.9D) compared to their input values (figure 7.9C). Nevertheless, except rectification, the nonlinear network amplifies the inMain strategies: Spiking vs Firingrecurrent rate models. • 2for put signal selectively in a similar manner as the linear network. • The issue of the emergence of orientation selectivity as a model A Recurrent of Simple in Primary Visual Cortex problem,Model extensively studiedCells theoretically and experimentally. - Two main models: feed-forward and recurrent. In chapter 2, we discussed feedforward model in which - Detailed spiking modelsahave been constructed which canthe be elongated directly receptive fields of simple cells in primary visual cortex were formed by compared to electrophysiology summing the inputs from lateral geniculate (LGN) neurons with their re- The same probleminisalternating also investigated with a firing rate cells. model,While a.k.a.this ceptive fields arranged rows of ON and OFF thequite !ringsuccessfully model". model accounts for a number of features of simple cells, such as orientation tuning, it is difficult to reconcile with the anatomy and circuitry of the cerebral cortex. By far the majority of the synapses onto any cortical neuron arise from other cortical neurons, not from thalamic afferents. Therefore, feedforward models account for the response properties of cortical neurons while ignoring the inputs that are numerically most prominent. The large number of intracortical connections suggests, instead, that recurrent circuitry might play an important role in shaping the responses of neurons in primary visual cortex. The Ring Model (2) + (7.36) The Ring Model (3) "#! • h is input, can be tuned (Hubel Wiesel h(θ) = c[1 − " + " ∗ cos(2θ)] / "$ "#' scenario) or very broadly tuned. 01!2 Ben-Yishai, Bar-Or, and Sompolinsky (1995) developed a model at the N neurons, withfor preferred angle, θi ,evenly distributed •other extreme, which recurrent connections are the primary determiners of orientation tuning. The model is similar in structure to the model between −π/2 and π/2 of equations 7.35 and 7.33, except that it includes a global inhibitory inter• Neurons receive thalamic inputs h. action. In addition, because orientation angles are defined over the range + recurrent connections, with excitatory weights between from −π/2 to π/2, rather than over the full 2π range, the cosine functions nearby cells and inhibitory cells that arebasic equation of the in the model have extra weights factors between of 2 in them. The further (mexican-hat model,apart as we implement profile) it, is ! % " π/2 " $ dv(θ) dθ # " " τr = −v(θ) + h (θ) + −λ0 + λ1 cos(2(θ − θ )) v(θ ) dt −π/2 π "% "#& "#% "#$ "/ !!" " !" ()*+,-.-*(,/! • The steady-state can be solved analytically. Model analyzed like a physical system. !" -('./* where v(θ) is the firing rate of a neuron with preferred orientation θ. " !!" The input to the model represents the orientation-tuned feedforward in!$"" put arising from ON-center and OFF-center LGN cells responding to an !$!" oriented image. As a function of preferred orientation, the input for an image!#"" with !!" orientation " !"angle & = 0 is %&'()*+*'%),! h (θ) = Ac (1 − ' + ' cos(2θ)) Draft: December 19, 2000 (7.37) Theoretical Neuroscience • Model achieves i) orientation selectivity; ii) contrast invariance of tuning, even if input is very broad. • The width of orientation selectivity depends on the shape of the mexican-hat, but is independent of the width of the input. • Symmetry breaking /Attractor dynamics. The Ring Model (4) Attractor Networks 7.4 Recurrent Networks •Attractor network : a network of neurons, usually recurrently connected, whose time 27 dynamics settle to a stable pattern. A B C 80 firing rate (Hz) 30 v (Hz) h (Hz) 60 20 10 40 20 • That pattern may be stationary (fixed points), time-varying (e.g. cyclic), or even 80 80% 40% 20% 10% 60 stochastic-looking (e.g., chaotic). • The particular pattern a network settles to is called its !attractor". 40 •The ring model is called a line (or ring) attractor network. Its stable states are also 20 sometimes referred to as !bump attractors". 0 -40 -20 0 20 θ (deg) 40 0 -40 -20 0 20 θ (deg) 40 0 180 200 220 240 Θ (deg) Figure 7.10: The effect of contrast on orientation tuning. A) The feedforward input as a function of preferred orientation. The four curves, from top to bottom, correspond to contrasts of 80%, 40%, 20%, and 10%. B) The output firing rates in response to different levels of contrast as a function of orientation preference. These are also the response tuning curves of a single neuron with preferred orientation zero. As in A, the four curves, from top to bottom, correspond to contrasts of 80%, 40%, 20%, and 10%. The recurrent model had λ0 = 7.3, λ1 = 11, A = 40 Hz, and " = 0.1. C) Tuning curves measure experimentally at four contrast levels as indicated in the legend. (C adapted from Sompolinsky and Shapley, 1997; based on data from Sclar and Freeman, 1982.) Point Attractor Line Attractor A Recurrent Model of Complex Cells in Primary Visual Cortex In the model of orientation tuning discussed in the previous section, recurrent amplification enhances selectivity. If the pattern of network connectivity amplifies nonselective rather than selective responses, recurrent interactions can also decrease selectivity. Recall from chapter 2 that neurons in the primary visual cortex are classified as simple or complex depending on their sensitivity to the spatial phase of a grating stimulus. Simple cells respond maximally when the spatial positioning of the light and dark regions of a grating matches the locations of the ON and OFF regions of their receptive fields. Complex cells do not have distinct ON and OFF regions in their receptive fields and respond to gratings of the appropriate orientation and spatial frequency relatively independently of where their light and dark stripes fall. In other words, complex cells are insensitive to spatial phase. Chance, Nelson, and Abbott (1999) showed that complex cell responses could be generated from simple cell responses by a recurrent network. As in chapter 2, we label spatial phase preferences by the angle φ. The feedforward input h (φ) in the model is set equal to the rectified response of a simple cell with preferred spatial phase φ (figure 7.11A). Each neuron in the network is labeled by the spatial phase preference of its feedforward input. The network neurons also receive recurrent input given by the weight function M (φ − φ" ) = λ1 /(2πρφ ) that is the same for all conDraft: December 19, 2000 • Model was tested with stimuli containing more than 1 orientation (crosses) •Model fails to distinguish angles separated by 30 Theoretical Neuroscience deg, overestimates larger angles •spurious attractors with noise 31 32 The Ring Model (5): Sustained Activity Network models - summary • If recurrent connections are strong enough, the pattern of population activity once established can become independent of the structure of the input. It can persists when input is removed. • A model of working memory ? • Network models: to understand the implications of connectivity in terms of computation and dynamics. • 2 Main strategies: Spiking vs Firing rate models. • The issue of the emergence of orientation selectivity as a model problem, extensively studied theoretically and experimentally. - Two main models: feed-forward and recurrent. - Detailed spiking models have been constructed which can be directly compared to electrophysiology - The same problem is also investigated with a firing rate model, a.k.a. the !ring model".