Supplementay mateial fo Inteaction of Feedfowad and Feedback Steams in Visual Cotex in a Fiing-Rate Model of Columna Computations Tobias Bosch and Heiko Neumann Institute fo Neual Infomation Pocessing Univesity of Ulm, James-Fanck-Ring, 89069 Ulm, Gemany Supplement : Absolute stability of the feedfowad/feedback cicuit model A cicuit mechanism has been developed that is capable to match physiological data of pyamidal cells with a uality compaable to a biophysically ealistic -compatment integate-and-fie model as developed in Lakum et al. (004). We have focused on a model cicuit that is based on gadual activation dynamics mimicking conductance-based changes in membane potential. The suggested cicuit components have been combined into a small netwok which consists of thee mutually coupled dynamic elements. The netwok is diven by feedfowad input signals p and a gain enhancing that is contolled by (modulating) feedback signals net. The stength of the intenal ecuent loop [g() ] is contolled by the paamete ε. The dynamical system is defined by τ τ! α +! + net ( β ) τ! α + p γ p δ + i + ε g ( ) + i ton ton with the theshold linea fiing ate function ( ) max( θ,0) g. Fo detailed analysis we assume that the inhibitoy stage uickly elaxes to euilibium such that the algebaic solution can be plugged into the euation of stage. This yields the - plana system F (, ) τ! α + ( β ) p δ + iton (, ) τ! α + p γ net + ε g( ) G ( ) + iton (S). (S) Fo values ε > 0 the system becomes a non-linea ecuent system and its stability needs to be analyzed. Hee, the esulting system can be efomulated as a competitive neual netwok, namely! x!! x! x x ( b ( x ) d( x )) ( b ( x ) d ( x )) (S3)
with the paticula esponse functions b b d d ( x ) ( x ) with ( ) max(,0) ( x ) δ g( x ) ( x ) γ ε g( x ) β p + i α + x p + i α + x ~ ton ton p γ net (S4) g ~ x x. This netwok is a special case of the geneal competitive netwok stuctues analyzed by Cohen & Gossbeg (983). Such netwoks ae absolutely stable given that cetain popeties ae fulfilled so that a global Liapunov function exists fo systems tajectoies which convege to sets with d / dt V( x) 0. We need to show that six citeia ae fulfilled such that the tajectoies ae guaanteed to convege to stable euilibium sets. a) The connection matix Λ (w ij ) fo the dynamic elements is symmetic and with positive coefficients w ij 0. b) The esponse functions a i (x i ) x i (i,) (en.s3) ae thus continuous fo all ξ fom 0 to x i. Fo the functions b i (x i ) we have two diffeent instances (en.s4). Given a diving input souce p 0 the function b (x ) is continuous and monotonically deceasing fo all inputs ξ > 0. Also fo b (x ) given p 0 and net 0 the function is continuous and monotonically deceasing fo all inputs ξ > 0. c) The esponse functions a i (x i ) x i (i,) (en.s3) ae thus continuous fo all ξ > 0. The functions in the competitive inteaction tems, d i (x j ) (en.s4), both fulfill the positivity constaint, d i (x j ) 0 fo all agument settings ξ. d) The competitive inteactions d i (x j ) ae both diffeentiable, given that g( ) is diffeentiable fo positive aguments, and the functions d i (x j ) ae both monotonically non-deceasing fo ξ 0 since g( ) is a monotonically inceasing fiing ate. e) The tajectoies fo the competitive inteaction should be descending, in paticula lim sup ξ (b i (ξ) - w ii d i (ξ)) < 0 fo all i,. In ou system we get β p + iton lim sup ~ ξ α + p δ g( ξ ) < 0 and ξ p + iton lim sup ξ α + γ net γ ε g < ξ ( ξ ) 0
given p 0 and net 0. f) Since b i (x i ) /x i (en.s4) the condition limξ 0+ b i (ξ) holds. This concludes the poof of stability fo the cicuit with the intenal ecuence [g() ]. Supplement : Feedback gain as a function of the stength of the intenal ecuence The euilibium esponse of the system as analyzed above leads to the steady-state euations ( α + p) ( α + γ net ) + ( γ ε β δ ) p i δ ) γ ε ( α + p) ( α + p) ( α + γ net ) ( γ ε β δ ) p iton δ ) δ ( α + γ net ) ton + Ξ, + Ξ. The tem Ξ is pesented in thee diffeent fomats in ode to highlight diffeent analytical popeties of Ξ in the following poofs. We have Ξ Ξ Ξ ( ( α + p) ( α + γ net ) + ( γ ε β δ ) p δ iton ) + 4 γ ε β p ( α + p) ( α + γ net ) ( α + p) ( α + γ net ) + ( γ ε β δ ) p δ iton ) + 4 δ ( α + p) ( α + γ net ) ( p + i ) ( α + p) ( α + γ net ) + ( γ ε β + δ ) p + δ iton ) 4 γ ε β δ p ( p + i ) ton, ton (S5), and. (S6) The fist two vaiants show that Ξ 0, i.e. Ξ is eal valued. The fist vaiant poves that 0 0 fo all vaiables being positive (which we geneally assume thoughout the pape). Vaiant two poves that Ξ is a monotonic inceasing function in ε if the tem in suae backets is positive, e.g. fo ε and/o net big enough. Fo ε, Ξ becomes. The thid vaiant will be used in the esults about the feedback sensitivity in the following. The feedback sensitivity is chaacteized by the ate of change of the steady-state solution depending on the stength of the feedback signal, net d 4 d net ε Ξ γ β p ε δ ( p + iton ) ( α + p) ( α + γ net ) + ( γ ε β + δ ) p + i δ + Ξ) ton (S7) 3
Using the thid vaiant of Ξ (en.s6), one can see that en.(s7) is always positive and that the ight-hand side becomes 0 fo ε. Vaiant of Ξ in en. (S6) poves that it is eal valued. This demonstates that fo lage enough values of ε and/o net input the ight-hand side of en.(s7) is monotonically deceasing such that fo lage values of ε the net feedback gain diminishes. This demonstates that stonge ecuence [g() ] leads to less modulatoy enhancement of the feeding input by the feedback signals. We hypothesize that cells in a cotical hypecolumn might show a statistical vaiation of the stength of ecuence such that some cells show the gain modulation via feedback while othes do not. It has been suggested that diffeent types of neuons exist, namely those that exhibit topdown attention selectivity (A-cells) and those cells that do not (N-cells; Roelfsema, 006). The esults deived hee make a suggestion of thei possible implementation at the netwok level. Supplement 3: Futhe simulations In ode to diectly compae the esponse chaacteistics of the detailed -compatment IF model (descibed in Lakum et al., 004) and the cicuit model poposed hee, we an compute simulations fo both model implementations. A pictoial oveview of the two models is pesented in Fig.S g() I m gate θ net h gate I S S p Figue S: Cicuit diagam of the -compatment model developed by Lakum et al. (004) (left) and the netwok level model poposed hee (ight). The components of the -compatment model eplicate the dynamic elements as summaized in ens.(a) and (b). The backpopagation mechanism is established by the signal flow fom the S to the compatment. The netwok level 4
cicuit diagam is eplicated fom Fig.(a) showing the majo signal pathways with diving input p and modulating feedback net and the excitatoy and inhibitoy inteactions. In ode to geneate a fiing-ate output fo the -compatment model we took the aveage spike ate in a fixed inteval as in (Lakum et al., 004). We adopted the 4- system as biefly eviewed in ens.(a) and (b) with the paametes taken fom (Lakum et al., 004). Additionally, we intoduced a scaling facto of the somatic input cuent I inj,s which was set to 0.7. Thee distal input stengths wee tested, namely 0 (no distal input), 50, and 750 pa (see Fig.S legend, top). This expeimental potocol is the same as fo geneating the esults in Fig.. Fo compaison, we eplicated Fig., bottom (simulation esults of the poposed cicuit with a ecuent intenal loop). Figue S: Simulation esults and fit of model esponses to physiological data epoted in Lakum et al. (004; thei Fig.5). The somatic (diving) input cuents ae inceased monotonically in the 5
ange of [0... na]. The fiing ate outputs ae geneated fo thee diffeent feedback signal levels of d ( net ) {0, 50, 750 pa}. Results ae shown fo the -compatmemt model (top) and the poposed cicuit model (bottom; Fig., main manuscipt). We see that both models fit the expeimental data. Fo the condition without distal (o feedback) input the esponses look almost identical. Fo non-zeo distal input the esponse chaacteistics slightly deviate. In the -compatment model output esponses tend to be geneated in a smooth esponse incease while the cicuit model shows shape tansitions. This is achieved by the suppessive effect of the tonic inhibition in the model neuon. The poposed model cicuit fits the data by diffeent steepness of the esponse fo the diffeent feedback signal levels. This is diectly eflected by the modulating action of the association of the two input steams. The poposed netwok model has been futhe analyzed computationally by systematically pobing the cicuit using combinations of signals with diffeent input stengths. This investigation focuses on the impact feedback signals net of diffeent stength have on the modulatoy gain of the diving input signal p. In the fist expeiment we chose an input of fixed amplitude and vaied the stength of the net feedback signal. The esults ae shown in Fig.S3. Figue S3: isplay of simulation esults of the poposed cicuit given a fixed amplitude diving feedfowad signal p. Feedback signals net have been augmented afte a shot delay (like in Fig.3, 6
main manuscipt). The amplitudes of these signals vay in discete steps net {0.5, 0.5, 0.75}. The stength of the intenal ecuence was set to zeo, ε 0. What can be seen fom these simulations is that the output gain vaies monotonically as a function of the feedback stength, so that d / d net [ ] 0. We note that in the limit fo net the esponse is diven towad the satuation level as defined by β. In a second expeiment we eplicated the expeiment shown in Fig. 3 (main manuscipt) with the paametes fitted fom Fig. (top). The esults ae shown in Fig.S4. By changing the value of ε {0.0, 0.0, 0.04} allows us to investigate the incease of the output stength (due to the monotonically deceasing self-inhibition of the diving input signal) and the eduction of the gain to amplify the output signal as a monotonic function of net. Figue S4: isplay of simulation esults of the poposed cicuit given fixed amplitude diving feedfowad and modulating feedback signals, p and net, espectively. 7
These simulations confim the theoetical pedictions that the gain enhancement by the modulating feedback net is educed in stength when the scaling of the ecuence [g() ] inceases. Refeences Cohen M.A., Gossbeg S. (983) Absolute stability of global patten fomation and paallel memoy stoage by competitive neual netwoks. IEEE Tans. on Systems, Man, and Cybenetics 3(5), 85-86 Lakum M.E., Senn W., Lüsche H.-R. (004) Top-down denditic input inceases the gain of laye 5 pyamidal neuons. Ceebal Cotex 4, 059-070 Roelfsema P.R. (006) Cotical algoithms fo peceptual gouping. Annual Review of Neuoscience 9, 03-7 8