{"title": "Emergence of Multiplication in a Biophysical Model of a Wide-Field Visual Neuron for Computing Object Approaches: Dynamics, Peaks, & Fits", "book": "Advances in Neural Information Processing Systems", "page_first": 469, "page_last": 477, "abstract": "Many species show avoidance reactions in response to looming object approaches. In locusts, the corresponding escape behavior correlates with the activity of the lobula giant movement detector (LGMD) neuron. During an object approach, its firing rate was reported to gradually increase until a peak is reached, and then it declines quickly. The $\\eta$-function predicts that the LGMD activity is a product between an exponential function of angular size $\\exp(-\\Theta)$ and angular velocity $\\dot{\\Theta}$, and that peak activity is reached before time-to-contact (ttc). The $\\eta$-function has become the prevailing LGMD model because it reproduces many experimental observations, and even experimental evidence for the multiplicative operation was reported. Several inconsistencies remain unresolved, though. Here we address these issues with a new model ($\\psi$-model), which explicitly connects $\\Theta$ and $\\dot{\\Theta}$ to biophysical quantities. The $\\psi$-model avoids biophysical problems associated with implementing $\\exp(\\cdot)$, implements the multiplicative operation of $\\eta$ via divisive inhibition, and explains why activity peaks could occur after ttc. It consistently predicts response features of the LGMD, and provides excellent fits to published experimental data, with goodness of fit measures comparable to corresponding fits with the $\\eta$-function.", "full_text": "Emergence of Multiplication in a Biophysical Model\nof a Wide-Field Visual Neuron for Computing Object\n\nApproaches: Dynamics, Peaks, & Fits\n\nMatthias S. Keil(cid:3)\n\nDepartment of Basic Psychology\n\nUniversity of Barcelona\nE-08035 Barcelona, Spain\n\nmatskeil@ub.edu\n\nAbstract\n\nMany species show avoidance reactions in response to looming object approaches.\nIn locusts, the corresponding escape behavior correlates with the activity of the\nlobula giant movement detector (LGMD) neuron. During an object approach, its\n\ufb01ring rate was reported to gradually increase until a peak is reached, and then\nit declines quickly. The (cid:17)-function predicts that the LGMD activity is a product\nbetween an exponential function of angular size exp((cid:0)(cid:2)) and angular velocity _(cid:2),\nand that peak activity is reached before time-to-contact (ttc). The (cid:17)-function has\nbecome the prevailing LGMD model because it reproduces many experimental\nobservations, and even experimental evidence for the multiplicative operation was\nreported. Several inconsistencies remain unresolved, though. Here we address\nthese issues with a new model ( -model), which explicitly connects (cid:2) and _(cid:2) to\nbiophysical quantities. The -model avoids biophysical problems associated with\nimplementing exp((cid:1)), implements the multiplicative operation of (cid:17) via divisive\ninhibition, and explains why activity peaks could occur after ttc. It consistently\npredicts response features of the LGMD, and provides excellent \ufb01ts to published\nexperimental data, with goodness of \ufb01t measures comparable to corresponding \ufb01ts\nwith the (cid:17)-function.\n\n1 Introduction: (cid:28) and (cid:17)\n\nCollision sensitive neurons were reported in species such different as monkeys [5, 4], pigeons\n[36, 34], frogs [16, 20], and insects [33, 26, 27, 10, 38]. This indicates a high ecological relevance,\nand raises the question about how neurons compute a signal that eventually triggers corresponding\nmovement patterns (e.g. escape behavior or interceptive actions). Here, we will focus on visual\nstimulation. Consider, for simplicity, a circular object (diameter 2l), which approaches the eye at\na collision course with constant velocity v. If we do not have any a priori knowledge about the\nobject in question (e.g. its typical size or speed), then we will be able to access only two information\nsources. These information sources can be measured at the retina and are called optical variables\n(OVs). The \ufb01rst is the visual angle (cid:2), which can be derived from the number of stimulated photore-\nceptors (spatial contrast). The second is its rate of change d(cid:2)(t)=dt (cid:17) _(cid:2)(t). Angular velocity _(cid:2) is\nrelated to temporal contrast.\nHow should we combine (cid:2) and _(cid:2) in order to track an imminent collision? The perhaps simplest\ncombination is (cid:28) (t) (cid:17) (cid:2)(t)= _(cid:2)(t) [13, 18]. If the object hit us at time tc, then (cid:28) (t) (cid:25) tc (cid:0) t will\nAlso: www.ir3c.ub.edu, Research Institute for Brain, Cognition, and Behaviour (IR3C) Edi\ufb01ci de\n\n(cid:3)\n\nPonent, Campus Mundet, Universitat de Barcelona, Passeig Vall d\u2019Hebron, 171. E-08035 Barcelona\n\n1\n\n\fgive us a running estimation of the time that is left until contact1. Moreover, we do not need to know\nanything about the approaching object: The ttc estimation computed by (cid:28) is practically independent\nof object size and velocity. Neurons with (cid:28)-like responses were indeed identi\ufb01ed in the nucleus re-\ntundus of the pigeon brain [34]. In humans, only fast interceptive actions seem to rely exclusively on\n(cid:28) [37, 35]. Accurate ttc estimation, however, seems to involve further mechanisms (rate of disparity\nchange [31]).\nAnother function of OVs with biological relevance is (cid:17) (cid:17) _(cid:2) exp((cid:0)(cid:11)(cid:2)), with (cid:11) = const: [10].\nWhile (cid:17)-type neurons were found again in pigeons [34] and bullfrogs [20], most data were gath-\nered from the LGMD2 in locusts (e.g. [10, 9, 7, 23]). The (cid:17)-function is a phenomenological model\nfor the LGMD, and implies three principal hypothesis: (i) An implementation of an exponential\nfunction exp((cid:1)). Exponentation is thought to take place in the LGMD axon, via active membrane\nconductances [8]. Experimental data, though, seem to favor a third-power law rather than exp((cid:1)).\n(ii) The LGMD carries out biophysical computations for implementing the multiplicative operation.\nIt has been suggested that multiplication is done within the LGMD itself, by subtracting the loga-\nrithmically encoded variables log _(cid:2) (cid:0) (cid:11)(cid:2) [10, 8]. (iii) The peak of the (cid:17)-function occurs before\nttc, at visual angle (cid:2)(^t) = 2 arctan(1=(cid:11)) [9]. It follows ttc for certain stimulus con\ufb01gurations (e.g.\nl=jvj / 5ms). In principle, ^t > tc can be accounted for by (cid:17)(t + (cid:14)) with a \ufb01xed delay (cid:14) < 0 (e.g.\n(cid:0)27ms). But other researchers observed that LGMD activity continuous to rise after ttc even for\nl=jvj \u2019 5ms [28]. These discrepancies remain unexplained so far [29], but stimulation dynamics\nperhaps plays a role.\nWe we will address these three issues by comparing the novel function \u201c \u201d with the (cid:17)-function.\n\n2 LGMD computations with the -function: No multiplication, no\n\nexponentiation\n\nA circular object which starts its approach at distance x0 and with speed v projects a visual angle\n(cid:2)(t) = 2 arctan[l=(x0 (cid:0) vt)] on the retina [34, 9]. The kinematics is hence entirely speci\ufb01ed by the\nhalf-size-to-velocity ratio l=jvj, and x0. Furthermore, _(cid:2)(t) = 2lv=((x0 (cid:0) vt)2 + l2).\nIn order to de\ufb01ne , we consider at \ufb01rst the LGMD neuron as an RC-circuit with membrane poten-\ntial3 V [17]\n\n= (cid:12) (Vrest (cid:0) V ) + gexc (Vexc (cid:0) V ) + ginh (Vinh (cid:0) V )\n\n(1)\nCm = membrane capacity4; (cid:12) (cid:17) 1=Rm denotes leakage conductance across the cell membrane\n(Rm: membrane resistance); gexc and ginh are excitatory and inhibitory inputs. Each conductance\ngi (i = exc; inh) can drive the membrane potential to its associated reversal potential Vi (usually\nVinh (cid:20) Vexc). Shunting inhibition means Vinh = Vrest. Shunting inhibition lurks \u201csilently\u201d because\nit gets effective only if the neuron is driven away from its resting potential. With synaptic input, the\nneuron decays into its equilibrium state\n\ndV\ndt\n\nCm\n\nV1 (cid:17) Vrest(cid:12) + Vexcgexc + Vinhginh\n\n(2)\naccording to V (t) = V1(1 (cid:0) exp((cid:0)t=(cid:28)m)). Without external input, V (t (cid:29) 1) ! Vrest. The\ntime scale is set by (cid:28)m. Without synaptic input (cid:28)m (cid:17) Cm=(cid:12). Slowly varying inputs gexc; ginh > 0\nmodify the time scale to approximately (cid:28)m=(1 + (gexc + ginh)=(cid:12)). For highly dynamic inputs, such\nas in late phase of the object approach, the time scale gets dynamical as well. The -model assigns\nsynaptic inputs5\n\n(cid:12) + gexc + ginh\n\ngexc(t) = _#(t);\nginh(t) = [(cid:13)#(t)]e ;\n\n_#(t (cid:0) (cid:1)tstim) + (1 (cid:0) (cid:16)1) _(cid:2)(t)\n_#(t) = (cid:16)1\n#(t) = (cid:16)0#(t (cid:0) (cid:1)tstim) + (1 (cid:0) (cid:16)0)(cid:2)(t)\n\n(3a)\n(3b)\n\n1This linear approximation gets worse with increasing (cid:2), but turns out to work well until short before ttc ((cid:28)\n\nadopts a minimum at tc (cid:0) 0:428978 (cid:1) l=jvj).\n\n2LGMD activity is usually monitored via its postsynaptic neuron, the Descending Contralateral Movement\nDetector (DCMD) neuron. This represents no problem as LGMD spikes follow DCMD spikes 1:1 under visual\nstimulation [22] from 300Hz [21] to at least 400Hz [24].\n\n3Here we assume that the membrane potential serves as a predictor for the LGMD\u2019s mean \ufb01ring rate.\n4Set to unity for all simulations.\n5LGMD receives also inhibition from a laterally acting network [21]. The (cid:17)-function considers only direct\n\nfeedforward inhibition [22, 6], and so do we.\n\n2\n\n\f(a) discretized optical variables\n\n(b) versus (cid:17)\n\nFigure 1: (a) The continuous visual angle of an approaching object is shown along with its dis-\ncretized version. Discretization transforms angular velocity from a continuous variable into a series\nof \u201cspikes\u201d (rescaled). (b) The function with the inputs shown in a, with nrelax = 25 relaxation\ntime steps. Its peak occurs tmax = 56ms before ttc (tc = 300ms). An (cid:17) function ((cid:11) = 3:29) that\nwas \ufb01tted to shows good agreement. For continuous optical variables, the peak would occur 4ms\nearlier, and (cid:17) would have (cid:11) = 4:44 with R2 = 1. For nrelax = 10, is farther away from its\nequilibrium at V1, and its peak moves 19ms closer to ttc.\n\n(a) different nrelax\n\n(b) different (cid:1)tstim\n\nFigure 2: The \ufb01gures plot the relative time tmax (cid:17) tc (cid:0) ^t of the response peak of , V (^t), as a\nfunction of half-size-to-velocity ratio (points). Line \ufb01ts with slope (cid:11) and intercept (cid:14) were added\n(lines). The predicted linear relationship in all cases is consistent with experimental evidence [9].\n(a) The stimulus time scale is held constant at (cid:1)tstim = 1ms, and several LGMD time scales\nare de\ufb01ned by nrelax (= number of intercalated relaxation steps for each integration time step).\nBigger values of nrelax move V (t) closer to its equilibrium V1(t), implying higher slopes (cid:11) in\nturn. (b) LGMD time scale is \ufb01xed at nrelax = 25, and (cid:1)tstim is manipulated. Because of the\ndiscretization of optical variables (OVs) in our simulation, increasing (cid:1)tstim translates to an overall\nsmaller number of jumps in OVs, but each with higher amplitude.\n\nThus, we say (t) (cid:17) V (t) if and only if gexc and ginh are de\ufb01ned with the last equation. The time\nscale of stimulation is de\ufb01ned by (cid:1)tstim (by default 1ms). The variables # and _# are lowpass \ufb01ltered\nangular size and rate of expansion, respectively. The amount of \ufb01ltering is de\ufb01ned by memory\nconstants (cid:16)0 and (cid:16)1 (no \ufb01ltering if zero). The idea is to continue with generating synaptic input\nafter ttc, where (cid:2)(t > tc) = const and thus _(cid:2)(t > tc) = 0. Inhibition is \ufb01rst weighted by (cid:13),\nand then potentiated by the exponent e. Hodgkin-Huxley potentiates gating variables n; m 2 [0; 1]\ninstead (potassium / n4, sodium / m3, [12]) and multiplies them with conductances. Gabbiani\nand co-workers found that the function which transforms membrane potential to \ufb01ring rate is better\ndescribed by a power function with e = 3 than by exp((cid:1)) (Figure 4d in [8]).\n\n3\n\n050100150200250300350100time [ms]log \u0398(t)\u0398 \u2208 [7.63\u00c2\u00b0, 180.00\u00c2\u00b0[ temporal resolution \u2206 tstim=1.0ms scaled d\u0398/dtcontinuousdiscretized050100150200250300350\u22120.01\u22120.00500.0050.010.0150.020.0250.030.0350.04time [ms]amplitude l/|v|=20.00ms, \u03b2=1.00, \u03b3=7.50, e=3.00, \u03b60=0.90, \u03b61=0.99, nrelax=25 \u0398(t) (input)\u03d1(t) (filtered)voltage V(t) (output)tmax= 56mstc=300ms\u03b7(t): \u03b1=3.29, R2=1.00nrelax=10 \u2192 tmax=37ms5101520253035404550\u221250050100150200250l/|v| [ms]tmax [ms]tc=500ms, dia=12.0cm, \u2206tstim=1.00ms, dt=10.00\u00b5s, discrete=1 \u03b2=1.00, \u03b3=7.50, e=3.00, Vinh=\u22120.001, \u03b60=0.90, \u03b61=0.99nrelax = 50\u03b1=4.66, R2=0.99 [normal]nrelax = 25\u03b1=3.91, R2=1.00 [normal]nrelax = 0\u03b1=1.15, R2=0.99 [normal]\f3 Dynamics of the -function\n\nDiscretization. In a typical experiment, a monitor is placed a short distance away from the insect\u2019s\neye, and an approaching object is displayed. Computer screens have a \ufb01xed spatial resolution, and\nas a consequence size increments of the displayed object proceed in discrete jumps. The locust\nretina is furthermore composed of a discrete array of ommatidia units. We therefore can expect\na corresponding step-wise increment of (cid:2) with time, although optical and neuronal \ufb01ltering may\nsmooth (cid:2) to some extent again, resulting in # (\ufb01gure 1). Discretization renders _(cid:2) discontinuous,\nwhat again will be alleviated in _#. For simulating the dynamics of , we discretized angular size\nwith (cid:13)oor((cid:2)), and _(cid:2)(t) (cid:25) [(cid:2)(t + (cid:1)tstim) (cid:0) (cid:2)(t)]=(cid:1)tstim. Discretized optical variables (OVs)\nwere re-normalized to match the range of original (i.e. continuous) OVs.\nTo peak, or not to peak? Rind & Simmons reject the hypothesis that the activity peak signals\nimpending collision on grounds of two arguments [28]: (i) If (cid:2)(t + (cid:1)tstim)(cid:0) (cid:2)(t) \u2019 3o in consec-\nutively displayed stimulus frames, the illusion of an object approach would be lost. Such stimulation\nwould rather be perceived as a sequence of rapidly appearing (but static) objects, causing reduced\nresponses. (ii) After the last stimulation frame has been displayed (that is (cid:2) = const), LGMD\nresponses keep on building up beyond ttc. This behavior clearly depends on l=jvj, also according\nto their own data (e.g. Figure 4 in [26]): Response build up after ttc is typically observed for suf\ufb01-\nciently small values of l=jvj. Input into in situations where (cid:2) = const and _(cid:2) = 0, respectively,\nis accommodated by # and _#, respectively.\nWe simulated (i) by setting (cid:1)tstim = 5ms, thus producing larger and more infrequent jumps in\ndiscrete OVs than with (cid:1)tstim = 1ms (default). As a consequence, #(t) grows more slowly (de-\nlayed build up of inhibition), and the peak occurs later (tmax (cid:17) tc (cid:0) ^t = 10ms with everything else\nidentical with \ufb01gure 1b). The peak amplitude ^V = V (^t) decreases nearly sixfold with respect to\ndefault. Our model thus predicts the reduced responses observed by Rind & Simmons [28].\nLinearity. Time of peak \ufb01ring rate is linearly related to l=jvj [10, 9]. The (cid:17)-function is consistent\nwith this experimental evidence: ^t = tc (cid:0) (cid:11)l=jvj + (cid:14) (e.g. (cid:11) = 4:7, (cid:14) = (cid:0)27ms). The -function\nreproduces this relationship as well (\ufb01gure 2), where (cid:11) depends critically on the time scale of bio-\nphysical processes in the LGMD. We studied the impact of this time scale by choosing 10(cid:22)s for the\nnumerical integration of equation 1 (algorithm: 4th order Runge-Kutta). Apart from improving the\nnumerical stability of the integration algorithm, is far from its equilibrium V1(t) in every moment\nt, given the stimulation time scale (cid:1)tstim = 1ms 6. Now, at each value of (cid:2)(t) and _(cid:2)(t), respec-\ntively, we intercalated nrelax iterations for integrating . Each iteration takes V (t) asymptotically\ncloser to V1(t), and limnrelax(cid:29)1 V (t) = V1(t). If the internal processes in the LGMD cannot keep\nup with stimulation (nrelax = 0), we obtain slopes values that underestimate experimentally found\nvalues (\ufb01gure 2a). In contrast, for nrelax \u2019 25 we get an excellent agreement with the experimen-\ntally determined (cid:11). This means that \u2013 under the reported experimental stimulation conditions (e.g.\n[9]) \u2013 the LGMD would operate relatively close to its steady state7.\nNow we \ufb01x nrelax at 25 and manipulate (cid:1)tstim instead (\ufb01gure 2b). The default value (cid:1)tstim = 1ms\ncorresponds to (cid:11) = 3:91. Slightly bigger values of (cid:1)tstim (2:5ms and 5ms) underestimate the ex-\nperimental (cid:11). In addition, the line \ufb01ts also return smaller intercept values then. We see tmax < 0 up\nto l=jvj (cid:25) 13:5ms \u2013 LGMD activity peaks after ttc! Or, in other words, LGMD activity continues\nto increase after ttc. In the limit, where stimulus dynamics is extremely fast, and LGMD processes\nare kept far from equilibrium at each instant of the approach, (cid:11) gets very small. As a consequence,\ntmax gets largely independent of l=jvj: The activity peak would cling to tmax although we varied\nl=jvj.\n\n4 Freeze! Experimental data versus steady state of \u201cpsi\u201d\n\nIn the previous section, experimentally plausible values for (cid:11) were obtained if is close to equilib-\nrium at each instant of time during stimulation. In this section we will thus introduce a steady-state\n\n6Assuming one (cid:1)tstim for each integration time step. This means that by default stimulation and biophys-\n\nical dynamics will proceed at identical time scales.\n\n7Notice that in this moment we can only make relative statements - we do not have data at hand for de\ufb01ning\n\nabsolute time scales\n\n4\n\n\f(a) (cid:12) varies\n\n(b) e varies\n\n(c) (cid:13) varies\n\nFigure 3: Each curve shows how the peak ^ 1 (cid:17) 1(^t) depends on the half-size-to-velocity ratio.\nIn each display, one parameter of 1 is varied (legend), while the others are held constant (\ufb01gure\ntitle). Line slopes vary according to parameter values. Symbol sizes are scaled according to rmse\n(see also \ufb01gure 4). Rmse was calculated between normalized 1(t) & normalized (cid:17)(t) (i.e. both\nfunctions 2 [0; 1] with original minimum and maximum indicated by the textbox). To this end, the\npeak of the (cid:17)-function was placed at tc, by choosing, at each parameter value, (cid:11) = jvj(cid:1) (tc (cid:0) ^t)=l (for\ndetermining correlation, the mean value of (cid:11) was taken across l=jvj).\n\n(a) (cid:12) varies\n\n(b) e varies\n\n(c) (cid:13) varies\n\nFigure 4: This \ufb01gure complements \ufb01gure 3.\nIt visualizes the time averaged absolute difference\nbetween normalized 1(t) & normalized (cid:17)(t). For (cid:17), its value of (cid:11) was chosen such that the\nmaxima of both functions coincide. Although not being a \ufb01t, it gives a rough estimate on how the\nshape of both curves deviate from each other. The maximum possible difference would be one.\n\nversion of (i.e. equation 2 with Vrest = 0, Vexc = 1, and equations 3 plugged in),\n\n 1(t) (cid:17) _(cid:2)(t) + Vinh [(cid:13)(cid:2)(t)]e\n(cid:12) + _(cid:2)(t) + [(cid:13)(cid:2)(t)]e\n\n(4)\n\n(Here we use continuous versions of angular size and rate of expansion). The 1-function\nmakes life easier when it comes to \ufb01tting experimental data. However, it has its limitations, be-\ncause we brushed the whole dynamic of under the carpet. Figure 3 illustrates how the lin-\near relationship (=\u201clinearity\u201d) between tmax (cid:17) tc (cid:0) ^t and l=jvj is in\ufb02uenced by changes in pa-\nrameter values. Changing any of the values of e, (cid:12), (cid:13) predominantly causes variation in line\nslopes. The smallest slope changes are obtained by varying Vinh (data not shown; we checked\nVinh = 0;(cid:0)0:001;(cid:0)0:01;(cid:0)0:1). For Vinh / (cid:0)0:01, linearity is getting slightly compromised, as\nslope increases with l=jvj (e.g. Vinh = (cid:0)1 (cid:11) 2 [4:2; 4:7]).\nIn order to get a notion about how well the shape of 1(t) matches (cid:17)(t), we computed time-\naveraged difference measures between normalized versions of both functions (details: \ufb01gure 3 & 4).\nBigger values of (cid:12) match (cid:17) better at smaller, but worse at bigger values of l=jvj (\ufb01gure 4a). Smaller\n(cid:12) cause less variation across l=jvj. As to variation of e, overall curve shapes seem to be best aligned\nwith e = 3 to e = 4 (\ufb01gure 4b). Furthermore, better matches between 1(t) and (cid:17)(t) correspond to\nbigger values of (cid:13) (\ufb01gure 4c). And \ufb01nally, Vinh marches again to a different tune (data not shown).\nVinh = (cid:0)0:1 leads to the best agreement ((cid:25) 0:04 across l=jvj) of all Vinh, quite different from the\nother considered values. For the rest, 1(t) and (cid:17)(t) align the same (all have maximum 0:094),\n\n5\n\n5101520253035404550050100150200250300l/|v| [ms]tmax [ms]tc=500ms, v=2.00m/s \u03c8\u221e \u2192 (\u03b2 varies), \u03b3=3.50, e=3.00, Vinh=\u22120.001 norm. |\u03b7\u2212\u03c8\u221e| = 0.020...0.128norm. rmse = 0.058...0.153correlation (\u03b2,\u03b1)=\u22120.90 (n=4)\u03b2=10.00\u03b2=5.00\u03b2=2.50\u03b2=1.005101520253035404550050100150200250300350l/|v| [ms]tmax [ms]tc=500ms, v=2.00m/s \u03c8\u221e \u2192 \u03b2=2.50, \u03b3=3.50, (e varies), Vinh=\u22120.001 norm. |\u03b7\u2212\u03c8\u221e| = 0.009...0.114norm. rmse = 0.014...0.160correlation (e,\u03b1)=0.98 (n=4)e=5.00e=4.00e=3.00e=2.505101520253035404550050100150200250300l/|v| [ms]tmax [ms]tc=500ms, v=2.00m/s \u03c8\u221e \u2192 \u03b2=2.50, (\u03b3 varies), e=3.00, Vinh=\u22120.001 norm. |\u03b7\u2212\u03c8\u221e| = 0.043...0.241norm. rmse = 0.085...0.315correlation (\u03b3,\u03b1)=1.00 (n=5)\u03b3=5.00\u03b3=2.50\u03b3=1.00\u03b3=0.50\u03b3=0.25510152025303540455000.020.040.060.080.10.120.14l/|v| [ms]meant |\u03b7(t)\u2212\u03c8\u221e(t)| (normalized \u03b7, \u03c8\u221e)tc=500ms, v=2.00m/s \u03c8\u221e \u2192 (\u03b2 varies), \u03b3=3.50, e=3.00, Vinh=\u22120.001 \u03b2=10.00\u03b2=5.00\u03b2=2.50\u03b2=1.001015202530354045500.020.040.060.080.10.12l/|v| [ms]meant |\u03b7(t)\u2212\u03c8\u221e(t)| (normalized \u03b7, \u03c8\u221e)tc=500ms, v=2.00m/s \u03c8\u221e \u2192 \u03b2=2.50, \u03b3=3.50, (e varies), Vinh=\u22120.001 e=5.00e=4.00e=3.00e=2.50510152025303540455000.050.10.150.20.25l/|v| [ms]meant |\u03b7(t)\u2212\u03c8\u221e(t)| (normalized \u03b7, \u03c8\u221e)tc=500ms, v=2.00m/s \u03c8\u221e \u2192 \u03b2=2.50, (\u03b3 varies), e=3.00, Vinh=\u22120.001 \u03b3=5.00\u03b3=2.50\u03b3=1.00\u03b3=0.50\u03b3=0.25\f(a) _(cid:2) = 126o=s\n\n(b) _(cid:2) = 63o=s\n\nFigure 5: The original data (legend label \u201cHaGaLa95\u201d) were resampled from ref. [10] and show\nDCMD responses to an object approach with _(cid:2) = const. Thus, (cid:2) increases linearly with time. The\n(cid:17)-function (\ufb01tting function: A(cid:17)(t+(cid:14))+o) and 1 (\ufb01tting function: A 1(t)+o) were \ufb01tted to these data:\n(a) (Figure 3 Di in [10]) Good \ufb01ts for 1 are obtained with e = 5 or higher (e = 3 R2 = 0:35 and\nrmse = 0:644; e = 4 R2 = 0:45 and rmse = 0:592). \u201cPsi\u201d adopts a sigmoid-like curve form which\n(subjectively) appears to \ufb01t the original data better than (cid:17). (b) (Figure 3 Dii in [10]) \u201cPsi\u201d yields an\nexcellent \ufb01t for e = 3.\n\n(a) spike trace\n\n(b) (cid:11) versus (cid:12)\n\n(a) DCMD activity in response to a black square (l=jvj = 30ms,\n\nFigure 6:\nlegend label\n\u201ce011pos14\u201d, ref. [30]) approaching to the eye center of a gregarious locust (\ufb01nal visual angle 50o).\nData show the \ufb01rst stimulation so habituation is minimal. The spike trace (sampled at 104Hz) was\nfull wave recti\ufb01ed, lowpass \ufb01ltered, and sub-sampled to 1ms resolution. Firing rate was estimated\nwith Savitzky-Golay \ufb01ltering (\u201csgolay\u201d). The \ufb01ts of the (cid:17)-function (A(cid:17)(t + (cid:14)) + o; 4 coef\ufb01cients) and\n 1-function (A 1(t) with \ufb01xed e; o; (cid:14); Vinh; 3 coef\ufb01cients) provide both excellent \ufb01ts to \ufb01ring rate.\n(b) Fitting coef\ufb01cient (cid:11) (! (cid:17)-function) inversely correlates with (cid:12) (! 1) when \ufb01tting \ufb01ring rates\nof another 5 trials as just described (continuous line = line \ufb01t to the data points). Similar correlation\nvalues would be obtained if e is \ufb01xed at values e = 2:5; 4; 5 c = (cid:0)0:95;(cid:0)0:96;(cid:0)0:91. If o was\ndetermined by the \ufb01tting algorithm, then c = (cid:0)0:70. No clear correlations with (cid:11) were obtained for\n(cid:13).\n\ndespite of covering different orders of magnitude with Vinh = 0;(cid:0)0:001;(cid:0)0:01.\nDecelerating approach. Hatsopoulos et al. [10] recorded DCMD activity in response to an ap-\nproaching object which projected image edges on the retina moving at constant velocity: _(cid:2) = const:\nimplies (cid:2)(t) = (cid:2)0 + _(cid:2)t. This \u201clinear approach\u201d is perceived as if the object is getting increasingly\nslower. But what appears a relatively unnatural movement pattern serves as a test for the functions\n(cid:17) & 1. Figure 5 illustrates that 1 passes the test, and consistently predicts that activity sharply\nrises in the initial approach phase, and subsequently declines ((cid:17) passed this test already in the year\n1995).\n\n6\n\n3.43.63.844.24.44.64.855.2time [s]RoHaTo10 gregarious locust LV=0.03s \u03c8\u221e: R2=0.95, rmse=0.004, 3 coefficients \u2192 \u03b2=2.22, \u03b3=0.70, e=3.00, Vinh=\u22120.001, A=0.07, o=0.02, \u03b4=0.00ms\u03b7: R2=1.00, rmse=0.001 \u2192 \u03b1=3.30, A=0.08, o=0.0, \u03b4=\u221210.5ms \u0398(t), lv=30mse011pos014sgolay with 100tmax=107msttc=5.00s \u03c8\u221e adj.R2 0.95 (LM:3) \u03b7(t) adj.R2 1 (TR::1)\fSpike traces. We re-sampled about 30 curves obtained from LGMD recordings from a variety of\npublications, and \ufb01tted (cid:17) & 1-functions. We cannot show the results here, but in terms of good-\nness of \ufb01t measures, both functions are in the same ballbark. Rather, \ufb01gure 6a shows a representative\nexample [30]. When (cid:11) and (cid:12) are plotted against each other for \ufb01ve trials, we see a strong inverse\ncorrelation (\ufb01gure 6b). Although \ufb01ve data points are by no means a \ufb01rm statistical sample, the\nstrong correlation could indicate that (cid:12) and (cid:11) play similar roles in both functions. Biophysically, (cid:12)\nis the leakage conductance, which determines the (passive) membrane time constant (cid:28)m / 1=(cid:12) of\nthe neuron. Voltage drops within (cid:28)m to exp((cid:0)1) times its initial value. Bigger values of (cid:12) mean\nshorter (cid:28)m (i.e., \u201cfaster neurons\u201d). Getting back to (cid:17), this would suggest (cid:11) / (cid:28)m, such that higher\n(absolute) values for (cid:11) would possibly indicate a slower dynamic of the underlying processes.\n\n5 Discussion (\u201cThe Good, the Bad, and the Ugly\u201d)\n\nUp to now, mainly two classes of LGMD models existed: The phenomenological (cid:17)-function on the\none hand, and computational models with neuronal layers presynaptic to the LGMD on the other\n(e.g. [25, 15]; real-world video sequences & robotics: e.g. [3, 14, 32, 2]). Computational models\npredict that LGMD response features originate from excitatory and inhibitory interactions in \u2013 and\nbetween \u2013 presynaptic neuronal layers. Put differently, non-linear operations are generated in the\npresynaptic network, and can be a function of many (model) parameters (e.g. synaptic weights, time\nconstants, etc.). In contrast, the (cid:17)-function assigns concrete nonlinear operations to the LGMD [7].\nThe (cid:17)-function is accessible to mathematical analysis, whereas computational models have to be\nprobed with videos or arti\ufb01cial stimulus sequences. The (cid:17)-function is vague about biophysical pa-\nrameters, whereas (good) computational models need to be precise at each (model) parameter value.\nThe (cid:17)-function establishes a clear link between physical stimulus attributes and LGMD activity: It\npostulates what is to be computed from the optical variables (OVs). But in computational models,\nsuch a clear understanding of LGMD inputs cannot always be expected: Presynaptic processing may\nstrongly transform OVs.\nThe function thus represents an intermediate model class: It takes OVs as input, and connects them\nwith biophysical parameters of the LGMD. For the neurophysiologist, the situation could hardly be\nany better. Psi implements the multiplicative operation of the (cid:17)-function by shunting inhibition\n(equation 1: Vexc (cid:25) Vrest and Vinh (cid:25) Vrest). The (cid:17)-function \ufb01ts very well according to our\ndynamical simulations (\ufb01gure 1), and satisfactory by the approximate criterion of \ufb01gure 4.\nWe can conclude that implements the (cid:17)-function in a biophysically plausible way. However, \ndoes neither explicitly specify (cid:17)\u2019s multiplicative operation, nor its exponential function exp((cid:1)). In-\nstead we have an interaction between shunting inhibition and a power law ((cid:1))e, with e (cid:25) 3. So what\nabout power laws in neurons?\nBecause of e > 1, we have an expansive nonlinearity. Expansive power-law nonlinearities are well\nestablished in phenomenological models of simple cells of the primate visual cortex [1, 11]. Such\nmodels approximate a simple cell\u2019s instantaneous \ufb01ring rate r from linear \ufb01ltering of a stimulus (say\nY ) by r / ([Y ]+)e, where [(cid:1)]+ sets all negative values to zero and lets all positive pass. Although\nexperimental evidence favors linear thresholding operations like r / [Y (cid:0) Ythres]+, neuronal re-\nsponses can behave according to power law functions if Y includes stimulus-independent noise [19].\nGiven this evidence, the power-law function of the inhibitory input into could possibly be inter-\npreted as a phenomenological description of presynaptic processes.\nThe power law would also be the critical feature by means of which the neurophysiologist could dis-\ntinguish between the (cid:17) function and . A study of Gabbiani et al. aimed to provide direct evidence\nfor a neuronal implementation of the (cid:17)-function [8]. Consequently, the study would be an evidence\nfor a biophysical implementation of \u201cdirect\u201d multiplication via log _(cid:2) (cid:0) (cid:11)(cid:2). Their experimental\nevidence fell somewhat short in the last part, where \u201cexponentation through active membrane con-\nductances\u201d should invert logarithmic encoding. Speci\ufb01cally, the authors observed that \u201cIn 7 out of\n10 neurons, a third-order power law best described the data\u201d (sixth-order in one animal). Alea iacta\nest.\n\nAcknowledgments\n\nMSK likes to thank Stephen M. Rogers for kindly providing the recording data for compiling \ufb01gure\n6. MSK furthermore acknowledges support from the Spanish Government, by the Ramon and Cajal\nprogram and the research grant DPI2010-21513.\n\n7\n\n\fReferences\n[1] D.G. Albrecht and D.B. Hamilton, Striate cortex of monkey and cat: contrast response function, Journal\n\nof Neurophysiology 48 (1982), 217\u2013237.\n\n[2] S. Bermudez i Badia, U. Bernardet, and P.F.M.J. Verschure, Non-linear neuronal responses as an emer-\ngent property of afferent networks: A case study of the locust lobula giant movemement detector, PLoS\nComputational Biology 6 (2010), no. 3, e1000701.\n\n[3] M. Blanchard, F.C. Rind, and F.M.J. Verschure, Collision avoidance using a model of locust LGMD\n\nneuron, Robotics and Autonomous Systems 30 (2000), 17\u201338.\n\n[4] D.F. Cooke and M.S.A. Graziano, Super-\ufb02inchers and nerves of steel: Defensive movements altered by\n\nchemical manipulation of a cortical motor area, Neuron 43 (2004), no. 4, 585\u2013593.\n\n[5] L. Fogassi, V. Gallese, L. Fadiga, G. Luppino, M. Matelli, and G. Rizzolatti, Coding of peripersonal space\n\nin inferior premotor cortex (area f4), Journal of Neurophysiology 76 (1996), 141\u2013157.\n\n[6] F. Gabbiani, I. Cohen, and G. Laurent, Time-dependent activation of feed-forward inhibition in a looming\n\nsensitive neuron, Journal of Neurophysiology 94 (2005), 2150\u20132161.\n\n[7] F. Gabbiani, H.G. Krapp, N. Hatsopolous, C.H. Mo, C. Koch, and G. Laurent, Multiplication and stimulus\n\ninvariance in a looming-sensitive neuron, Journal of Physiology - Paris 98 (2004), 19\u201334.\n\n[8] F. Gabbiani, H.G. Krapp, C. Koch, and G. Laurent, Multiplicative computation in a visual neuron sensitive\n\nto looming, Nature 420 (2002), 320\u2013324.\n\n[9] F. Gabbiani, H.G. Krapp, and G. Laurent, Computation of object approach by a wide-\ufb01eld, motion-\n\nsensitive neuron, Journal of Neuroscience 19 (1999), no. 3, 1122\u20131141.\n\n[10] N. Hatsopoulos, F. Gabbiani, and G. Laurent, Elementary computation of object approach by a wide-\ufb01eld\n\nvisual neuron, Science 270 (1995), 1000\u20131003.\n\n[11] D.J. Heeger, Modeling simple-cell direction selectivity with normalized, half-squared, linear operators,\n\nJournal of Neurophysiology 70 (1993), 1885\u20131898.\n\n[12] A.L. Hodkin and A.F. Huxley, A quantitative description of membrane current and its application to\n\nconduction and excitation in nerve, Journal of Physiology 117 (1952), 500\u2013544.\n\n[13] F. Hoyle, The black cloud, Pinguin Books, London, 1957.\n[14] M.S. Keil, E. Roca-Morena, and A. Rodr\u00b4\u0131guez-V\u00b4azquez, A neural model of the locust visual system for\ndetection of object approaches with real-world scenes, Proceedings of the Fourth IASTED International\nConference (Marbella, Spain), vol. 5119, 6-8 September 2004, pp. 340\u2013345.\n\n[15] M.S. Keil and A. Rodr\u00b4\u0131guez-V\u00b4azquez, Towards a computational approach for collision avoidance with\nreal-world scenes, Proceedings of SPIE: Bioengineered and Bioinspired Systems (Maspalomas, Gran\nCanaria, Canary Islands, Spain) (A. Rodr\u00b4\u0131guez-V\u00b4azquez, D. Abbot, and R. Carmona, eds.), vol. 5119,\nSPIE - The International Society for Optical Engineering, 19-21 May 2003, pp. 285\u2013296.\n\n[16] J.G. King, J.Y. Lettvin, and E.R. Gruberg, Selective, unilateral, reversible loss of behavioral responses\nto looming stimuli after injection of tetrodotoxin or cadmium chloride into the frog optic nerve, Brain\nResearch 841 (1999), no. 1-2, 20\u201326.\n\n[17] C. Koch, Biophysics of computation: information processing in single neurons, Oxford University Press,\n\nNew York, 1999.\n\n[18] D.N. Lee, A theory of visual control of braking based on information about time-to-collision, Perception\n\n5 (1976), 437\u2013459.\n\n[19] K.D. Miller and T.W. Troyer, Neural noise can explain expansive, power-law nonlinearities in neuronal\n\nresponse functions, Journal of Neurophysiology 87 (2002), 653\u2013659.\n\n[20] Hideki Nakagawa and Kang Hongjian, Collision-sensitive neurons in the optic tectum of the bullfrog, rana\n\ncatesbeiana, Journal of Neurophysiology 104 (2010), no. 5, 2487\u20132499.\n\n[21] M. O\u2019Shea and C.H.F. Rowell, Projection from habituation by lateral inhibition, Nature 254 (1975), 53\u2013\n\n55.\n\n[22] M. O\u2019Shea and J.L.D. Williams, The anatomy and output connection of a locust visual interneurone: the\nlobula giant movement detector (lgmd) neurone, Journal of Comparative Physiology 91 (1974), 257\u2013266.\n[23] S. Peron and F. Gabbiani, Spike frequency adaptation mediates looming stimulus selectivity, Nature Neu-\n\nroscience 12 (2009), no. 3, 318\u2013326.\n\n[24] F.C. Rind, A chemical synapse between two motion detecting neurones in the locust brain, Journal of\n\nExperimental Biology 110 (1984), 143\u2013167.\n\n[25] F.C. Rind and D.I. Bramwell, Neural network based on the input organization of an identi\ufb01ed neuron\n\nsignaling implending collision, Journal of Neurophysiology 75 (1996), no. 3, 967\u2013985.\n\n8\n\n\f[26] F.C. Rind and P.J. Simmons, Orthopteran DCMD neuron: a reevaluation of responses to moving objects.\nI. Selective responses to approaching objects, Journal of Neurophysiology 68 (1992), no. 5, 1654\u20131666.\n, Orthopteran DCMD neuron: a reevaluation of responses to moving objects. II. Critical cues for\n\n[27]\n\ndetecting approaching objects, Journal of Neurophysiology 68 (1992), no. 5, 1667\u20131682.\n\n[28]\n\n, Signaling of object approach by the dcmd neuron of the locust, Journal of Neurophysiology 77\n\n(1997), 1029\u20131033.\n\n, Reply, Trends in Neuroscience 22 (1999), no. 5, 438.\n\n[29]\n[30] S.M. Roger, G.W.J. Harston, F. Kilburn-Toppin, T. Matheson, M. Burrows, F. Gabbiani, and H.G. Krapp,\nSpatiotemporal receptive \ufb01eld properties of a looming-sensitive neuron in solitarious and gregarious\nphases of desert locust, Journal of Neurophysiology 103 (2010), 779\u2013792.\n\n[31] S.K. Rushton and J.P. Wann, Weighted combination of size and disparity: a computational model for\n\ntiming ball catch, Nature Neuroscience 2 (1999), no. 2, 186\u2013190.\n\n[32] Yue. S., Rind. F.C., M.S. Keil, J. Cuadri, and R. Stafford, A bio-inspired visual collision detection mech-\nanism for cars: Optimisation of a model of a locust neuron to a novel environment, Neurocomputing 69\n(2006), 1591\u20131598.\n\n[33] G.R. Schlotterer, Response of the locust descending movement detector neuron to rapidly approaching\n\nand withdrawing visual stimuli, Canadian Journal of Zoology 55 (1977), 1372\u20131376.\n\n[34] H. Sun and B.J. Frost, Computation of different optical variables of looming objects in pigeon nucleus\n\nrotundus neurons, Nature Neuroscience 1 (1998), no. 4, 296\u2013303.\n\n[35] J.R. Tresilian, Visually timed action: time-out for \u2019tau\u2019?, Trends in Cognitive Sciences 3 (1999), no. 8,\n\n1999.\n\n[36] Y. Wang and B.J. Frost, Time to collision is signalled by neurons in the nucleus rotundus of pigeons,\n\nNature 356 (1992), 236\u2013238.\n\n[37] J.P. Wann, Anticipating arrival: is the tau-margin a specious theory?, Journal of Experimental Psychol-\n\nogy and Human Perceptual Performance 22 (1979), 1031\u20131048.\n\n[38] M. Wicklein and N.J. Strausfeld, Organization and signi\ufb01cance of neurons that detect change of visual\ndepth in the hawk moth manduca sexta, The Journal of Comparative Neurology 424 (2000), no. 2, 356\u2013\n376.\n\n9\n\n\f", "award": [], "sourceid": 348, "authors": [{"given_name": "Matthias", "family_name": "Keil", "institution": null}]}