{"title": "An in-silico Neural Model of Dynamic Routing through Neuronal Coherence", "book": "Advances in Neural Information Processing Systems", "page_first": 1401, "page_last": 1408, "abstract": null, "full_text": "An in-silico Neural Model of Dynamic Routing\n\nthrough Neuronal Coherence\n\nDevarajan Sridharan\u2217\u2020, Brian Percival\u2217\u2021, John Arthur\\ and Kwabena Boahen\\\n\n\u2020 Program in Neurosciences,\n\n\u2021 Department of Electrical Engineering\n\nand \\ Department of Bioengineering\n\nStanford University\n\n\u2217 These authors contributed equally\n\n{dsridhar, bperci, jarthur, boahen}@stanford.edu\n\nAbstract\n\nWe describe a neurobiologically plausible model to implement dynamic routing\nusing the concept of neuronal communication through neuronal coherence. The\nmodel has a three-tier architecture: a raw input tier, a routing control tier, and an\ninvariant output tier. The correct mapping between input and output tiers is re-\nalized by an appropriate alignment of the phases of their respective background\noscillations by the routing control units. We present an example architecture, im-\nplemented on a neuromorphic chip, that is able to achieve circular-shift invariance.\nA simple extension to our model can accomplish circular-shift dynamic routing\nwith only O(N) connections, compared to O(N 2) connections required by tradi-\ntional models.\n\n1 Dynamic Routing Circuit Models for Circular-Shift Invariance\n\nDynamic routing circuit models are among the most prominent neural models for invariant recogni-\ntion [1] (also see [2] for review). These models implement shift invariance by dynamically changing\nspatial connectivity to transform an object to a standard position or orientation. The connectivity\nbetween the raw input and invariant output layers is controlled by routing units, which turn certain\nsubsets of connections on or off (Figure 1A). An important feature of this model is the explicit rep-\nresentation of what and where information in the main network and the routing units, respectively;\nthe routing units use the where information to create invariant representations.\n\nTraditional solutions for shift invariance are neurobiologically implausible for at least two reasons.\nFirst, there are too many synaptic connections: for N input neurons, N output neurons and N\npossible input-output mappings, the network requires O(N 2) connections in the routing layer\u2014\nbetween each of the N routing units and each set of N connections that that routing unit gates (Figure\n1A). Second, these connections must be extremely precise: each routing unit must activate an input-\noutput mapping (N individual connections) corresponding to the desired shift (as highlighted in\nFigure 1A). Other approaches that have been proposed, including invariant feature networks [3,4],\nalso suffer from signi\ufb01cant drawbacks, such as the inability to explicitly represent where information\n[2]. It remains an open question how biology could achieve shift invariance without pro\ufb02igate and\nprecise connections.\n\nIn this article, we propose a simple solution for shift invariance for quantities that are circular or\nperiodic in nature\u2014circular-shift invariance (CSI)\u2014orientation invariance in vision and key invari-\nance in music. The visual system may create orientation-invariant representations to aid recognition\nunder conditions of object rotation or head-tilt [5,6]; a similar mechanism could be employed by\nthe auditory system to create key-invariant representations under conditions where the same melody\n\n1\n\n\fFigure 1: Dynamic routing. A In traditional dynamic routing, connections from the (raw) input layer\nto the (invariant) output layer are gated by routing units. For instance, the mapping from A to 5, B to\n6, . . . , F to 4 is achieved by turning on the highlighted routing unit. B In time-division multiplexing\n(TDM), the encoder samples input channels periodically (using a rotating switch) while the decoder\nsends each sample to the appropriate output channel (based on its time bin). TDM can be extended to\nachieve a circular-shift transformation by altering the angle between encoder and decoder switches\n(\u03b8), thereby creating a rotated mapping between input and output channels (adapted from [7]).\n\nis played in different keys. Similar to orientation, which is a periodic quantity, musical notes one\noctave apart sound alike, a phenomenon known as octave equivalence [8]. Thus, the problems of\nkey invariance and orientation invariance admit similar solutions.\n\nDeriving inspiration from time-division multiplexing (TDM), we propose a neural network for CSI\nthat uses phase to encode and decode information. We modulate the temporal window of commu-\nnication between (raw) input and (invariant) output neurons to achieve the appropriate input\u2013output\nmapping. Extending TDM, any particular circular-shift transformation can be accomplished by\nchanging the relative angle, \u03b8, between the rotating switches of the encoder (that encodes the raw\ninput in time) and decoder (that decodes the invariant output in time) (Figure 1B). This obviates the\nneed to hardwire routing control units that speci\ufb01cally modulate the strength of each possible input-\noutput connection, thereby signi\ufb01cantly reducing the complexity inherent in the traditional dynamic\nrouting solution. Similarly, a remapping between the input and output neurons can be achieved by\nintroducing a relative phase-shift in their background oscillations.\n\n2 Dynamic Routing through Neuronal Coherence\n\nTo modulate the temporal window of communication, the model uses a ring of neurons (the oscilla-\ntion ring) to select the pool of neurons (in the projection ring) that encode or decode information at a\nparticular time (Figure 2A). Each projection pool encodes a speci\ufb01c value of the feature (for exam-\nple, one of twelve musical notes). Upon activation by external input, each pool is active only when\nbackground inhibition generated by the oscillation ring (outer ring of neurons) is at a minimum. In\naddition to exciting 12 inhibitory interneurons in the projection ring, each oscillation ring neuron\nexcites its nearest 18 neighbors in the clockwise direction around the oscillation ring. As a result, a\nwave of inhibition travels around the projection ring that allows only one pool to be excitable at any\npoint in time. These neurons become excitable at roughly the same time (numbered sectors, inner\nring) by virtue of recurrent excitatory intra-pool connections.\n\nDecoding is accomplished by a second tier of rings (Figure 2B). The projection ring of the \ufb01rst (in-\nput) tier connects all-to-all to the projection ring of the second (output) tier. The two oscillation rings\ncreate a window of excitability for the pools of neurons in their respective projection rings. Hence,\nthe most effective communication occurs between input and output pools that become excitable at\nthe same time (i.e. are oscillating in phase with one another [9]).\n\nThe CSI problem is solved by introducing a phase-shift between the input and output tiers. If they\nare exactly in phase, then an input pool is simply mapped to the output pool directly above it. If their\n\n2\n\n\fFigure 2: Double-Ring Network for Encoding and Decoding. A The projection (inner) ring is\ndivided into (numbered) pools. The oscillation (outer) ring modulates sub-threshold activity (wave-\nforms) of the projection ring by exciting (black distribution) inhibitory neurons that inhibit neigh-\nboring projection neurons. A wave of activity travels around the oscillation ring due to asymmetric\nexcitatory connections, creating a corresponding wave of inhibitory activity in the projection ring,\nsuch that only one pool of projection neurons is excitable (spikes) at a given time. B Two instances\nof the double-ring structure from A. The input projection ring connects all-to-all to the output pro-\njection ring (dashed lines). Because each input pool will spike only during a distinct time bin, and\neach output pool is excitable only in a certain time bin, communication occurs between input and\noutput pools that are oscillating in phase with each other. Appropriate phase offset between input\nand output oscillation rings realizes the desired circular shift (input pool H to output pool 1, solid\narrow). C Interactions among pools highlighted in B.\n\nphases are different, the input is dynamically routed to an appropriate circularly shifted position in\nthe output tier. Such changes in phase are analogous to adjusting the angle of the rotating switch\nat either the encoder or the decoder in TDM (see Figure 1B). There is some evidence that neural\nsystems could employ phase relationships of subthreshold oscillations to selectively target neural\npopulations [9-11].\n\n3 Implementation in Silicon\n\nWe implemented this solution to CSI on a neuromorphic silicon chip [12]. The neuromorphic chip\nhas neurons whose properties resemble that of biological neurons; these neurons even have intrin-\nsic differences, thereby mimicking heterogeneity in real neurobiological systems. The chip uses a\nconductance-based spiking model for both inhibitory and excitatory neurons. Inhibitory neurons\nproject to nearby excitatory and inhibitory neurons via a diffusor network that determines the spread\nof inhibition. A lookup table of excitatory synaptic connectivity is stored in a separate random-\naccess memory (RAM) chip. Spikes occurring on-chip are converted to a neuron address, mapped\nto synapses (if any) via the lookup table, and routed to the targeted on-chip synapse. A universal\nserial bus (USB) interface chip communicates spikes to and from a computer, for external input and\n\n3\n\n\fFigure 3: Traveling-wave activity in the oscillation ring. A Population activity (5ms bins) of a pool\nof eighteen (adjacent) oscillation neurons. B Increasing the strength of feedforward excitation led\nto increasing frequencies of periodic \ufb01ring in the \u03b8 and \u03b1 range (1-10 Hz). Strength of excitation\nis the amplitude change in post-synaptic conductance due to a single pre-synaptic spike (measured\nrelative to minimum amplitude used).\n\ndata analysis, respectively. Simulations on the chip occur in real-time, making it an attractive option\nfor implementing the model.\n\nWe con\ufb01gured the following parameters:\n\n\u2022 Magnitude of a potassium M-current: increasing this current\u2019s magnitude increased the\npost-spike repolarization time of the membrane potential, thereby constraining spiking to a\nsingle time bin per cycle.\n\n\u2022 The strength of excitatory and inhibitory synapses: a correct balance had to be established\nbetween excitation and inhibition to make only a small subset of neurons in the projection\nrings \ufb01re at a time\u2014too much excitation led to widespread \ufb01ring and too much inhibition\nled to neurons that were entirely silent or \ufb01red sporadically.\n\n\u2022 The space constant of inhibitory spread: increasing the spread was effective in preventing\n\nrunaway excitation, which could occur due to the recurrent excitatory connections.\n\nWe were able to create a stable traveling wave of background activity within the oscillation ring.\nWe transiently stimulated a small subset of the neurons, which initiated a wave of activity that\npropagated in a stable manner around the ring after the transient external stimulation had ceased\n(Figure 3A). The network frequency determined from a Fourier transform of the network activity\nsmoothed with a non-causal Gaussian kernel (FDHM = 80ms) was 7.4Hz. The frequency varied\nwith the strength of the neurons\u2019 excitatory connections (Figure 3B), measured as the amplitude of\nthe step increase in membrane conductivity due to the arrival of a pre-synaptic spike. Over much\nof the range of the synaptic strengths tested, we observed stable oscillations in the \u03b8 and \u03b1 bands\n(1-10Hz); the frequency appeared to increase logarithmically with synaptic strength.\n\n4 Phase-based Encoding and Decoding\n\nIn order to assess the best-case performance of the model, the background activity in the input and\noutput projection rings was derived from the input oscillation ring. Their spikes were delivered to\nthe appropriately circularly-shifted output oscillation neurons. The asymmetric feedforward con-\nnections were disabled in the output oscillation ring. For instance, in order to achieve a circular shift\nby k pools (i.e. mapping input projection pool 1 to output projection pool k + 1, input pool 2 to\noutput pool k + 2, and so on), activity from the input oscillation neurons closest to input pool 1 was\nfed into the output oscillation neurons closest to output pool k. By providing the appropriate phase\ndifference between input and output oscillation, we were able to assess the performance of the model\nunder ideal conditions. In the Discussion section, we discuss a biologically plausible mechanism to\ncontrol the relative phases.\n\n4\n\n\fFigure 4: Phase-based encoding. Rasters indicating activity of projection pools in 1ms bins, and\nmean phase of \ufb01ring (\u00d7\u2019s) for each pool (relative to arbitrary zero time). The abscissa shows \ufb01ring\ntime normalized by the period of oscillation (which may be converted to \ufb01ring phase by multiplica-\ntion by 2\u03c0). Under constant input to the input projection ring, the input pools \ufb01re approximately in\nsequence. Two cycles of pool activity normalized by maximum \ufb01ring rate for each pool are shown\nin left inset (for clarity, pools 1-6 are shown in the top panel and pools 7-12 are shown separately\nin the bottom panel); phase of background inhibition of pool 4 is shown (below) for reference.\nPhase-aligned average1 of activity (right inset) showed that the \ufb01ring times were relatively tight and\nuniform across pools: a standard deviation of 0.0945 periods, or equivalently, a spread of 1.135\npools at any instant of time.\n\nWe veri\ufb01ed that the input projection pools \ufb01red in a phase-shifted fashion relative to one another,\na property critical for accurate encoding (see Figure 2). We stimulated all pools in the input pro-\njection ring simultaneously while the input oscillation ring provided a periodic wave of background\ninhibition. The mean phase of \ufb01ring for each pool (relative to arbitrary zero time) increased nearly\nlinearly with pool number, thereby providing evidence for accurate, phase-based encoding (Figure\n4). The \ufb01ring times of all pools are shown for two cycles of background oscillatory activity (Figure 4\nleft inset). A phase-aligned average1 showed that the timing was relatively tight (standard deviation\n1.135 pools) and uniform across pools of neurons (Figure 4 right inset).\n\nWe then characterized the system\u2019s ability to correctly decode this encoding under a given circular\nshift. The shift was set to seven pools, mapping input pool 1 to output pool 8, and so on. Each input\npool was stimulated in turn. We expected to see only the appropriately shifted output pool become\nhighly active.\nIn fact, not only was this pool active, but other pools around it were also active,\nthough to a lesser extent (Figure 5A). Thus, the phase-encoded input was decoded successfully, and\ncircularly shifted, except that the output units were broadly tuned.\n\nTo quantify the overall precision of encoding and decoding, we constructed an input-locked aver-\nage of the tuning curves (Figure 5B): the curves were circularly shifted to the left by an amount\ncorresponding to the stimulated input pool number, and the raw pool \ufb01ring rates were averaged. If\nthe phase-based encoding and decoding were perfect, the peak should occur at a shift of 7 pools.\n\n1The phase-aligned average was constructed by shifting the pool-activity curves by the (# of the pool) \u00d7\n\n12 of the period) to align activity across pools, which was then averaged.\n( 1\n\n5\n\n\fFigure 5: Decoding phase-encoded input. A In order to assess decoding performance under a given\ncircular shift (here 7 pools) each input pool was stimulated in turn and activity in each output pool\nwas recorded and averaged over 500ms. The pool\u2019s response, normalized by its maximum \ufb01ring\nrate, is plotted for each stimulated input pool (arrows pointing to curves, color code as in Figure 4).\nEach input pool stimulation trial consistently resulted in peak activity in the appropriate output pool;\nhowever, adjacent pools were also active, but to a lesser extent, resulting in a broad tuning curve. B\nThe best-\ufb01t Gaussian (dot-dashed grey curve, \u03c3 = 2.30 pools) to the input-locked average of the raw\npool \ufb01ring rates (see text for details) revealed a maximum between a shift of 7 and 8 pools (inverted\ngrey triangle; expected peak at a shift of 7 pools).\n\nIndeed, the highest (average) \ufb01ring rate corresponded to a shift of 7 pools. However, the activity\ncorresponding to a shift of 8 pools was nearly equal to that of 7 pools, and the best \ufb01tting Gaus-\nsian curve to the activity histogram (grey dot-dashed line) peaked at a point between pools 7 and 8\n(inverted grey triangle). The standard deviation (\u03c3) was 2.30 pools, versus the expected ideal \u03c3 of\n1.60, which corresponds to the encoding distribution (\u03c3 = 1.135 pools) convolved with itself.\n\n5 Discussion\n\nWe have demonstrated a biologically plausible mechanism for the dynamic routing of information\nin time that obviates the need for precise gating of connections. This mechanism requires that a\nwave of activity propagate around pools of neurons arranged in a ring. While previous work has\ndescribed traveling waves in a ring of neurons [13], and a double ring architecture (for determining\nhead-direction) [14], our work combines these two features (twin rings with phase-shifted traveling\nwaves) to achieve dynamic routing. These features of the model are found in the cortex: Bonhoeffer\nand Grinwald [15] describe iso-orientation columns in the cat visual cortex that are arranged in\nring-like pinwheel patterns, with orientation tuning changing gradually around the pinwheel center.\nMoreover, Rubino et al. [16] have shown that coherent oscillations can propagate as waves across\nthe cortical surface in the motor cortex of awake, behaving monkeys performing a delayed reaching\ntask.\n\nOur solution for CSI is also applicable to music perception. In the Western twelve-tone, equal-\ntemperament tuning system (12-tone scale), each octave is divided into twelve logarithmically-\nspaced notes. Human observers are known to construct mental representations for raw notes that\nare invariant of the (perceived) key of the music: a note of C heard in the key of C-Major is percep-\ntually equivalent to the note C# heard in the key of C#-Major [8,17]. In previous dynamic routing\nmodels of key invariance, the tonic\u2014the \ufb01rst note of the key (e.g., C is the tonic of C-Major)\u2014\nsupplies the equivalent where information used by routing units that gate precise connections to\nmap the raw note into a key-invariant output representation [17].\n\nTo achieve key invariance in our model, the bottom tier encodes raw note information while the top\ntier decodes key-invariant notes (Figure 6). The middle tier receives the tonic information and aligns\nthe phase of the \ufb01rst output pool (whose invariant representation corresponds to the tonic) with the\nappropriate input pool (whose raw note representation corresponds to the tonic of the perceived key).\n\n6\n\n\fFigure 6: Phase-based dynamic routing to achieve key-invariance. The input (bottom) tier encodes\nraw note information, and the output (top) tier decodes key-invariant information. The routing\n(middle) tier sets the phase of the background wave activity in the input and output oscillation rings\n(dashed arrows) such that the \ufb01rst output pool is in phase with the input pool representing the note\ncorresponding to the tonic. On the left, where G is the tonic, input pool G, output pool 1, and the\nrouting tier are in phase with one another (black clocks), while input pool C and output pool 6 are in\nphase with one another (grey clocks). Thus, the raw note input, G, activates the invariant output 1,\nwhich corresponds to the perceived tonic invariant representation (heavy solid arrows). On the right,\nthe same raw input note, G, is active, but the key is different and A is now the active tonic; thus the\nraw input, G, is now mapped to output pool 11.\n\nThe tonic information is supplied to a speci\ufb01c pool in the routing ring according to the perceived\nkey. This pool projects directly down to the input pool corresponding to the tonic. This ensures\nthat the current tonic\u2019s input pool is excitable in the same time bin as the \ufb01rst output pool. Each\nof the remaining raw input notes of the octave is mapped by time binning to the corresponding\nkey-invariant representation in the output tier, as the phases of input pools are all shifted by the\nsame amount. Supporting evidence for phase-based encoding of note information comes from MEG\nrecordings in humans: the phase of the MEG signal (predominantly over right hemispheric sensor\nlocations) tracks the note of the heard note sequence with surprising accuracy [18].\n\nThe input and output tiers\u2019 periods must be kept in lock-step, which can be accomplished through\nmore plausible means than employed in the current implementation of this model. Here, we main-\ntained a \ufb01xed phase shift between the input and output oscillation rings by feeding activity from the\ninput oscillation ring to the appropriately shifted pool in the output oscillation ring. This approach\nallowed us to avoid dif\ufb01culties achieving coherent oscillations at identical frequencies in the input\nand output oscillation rings. Alternatively, entrainment could be achieved even when the frequencies\nare not identical\u2014a more biologically plausible scenario\u2014if the routing ring resets the phase of the\ninput and output rings on a cycle-by-cycle basis. Lakatos et al. [19] have shown that somatosen-\nsory inputs can reset the phase of ongoing neuronal oscillations in the primary auditory cortex (A1),\nwhich helps in the generation of a uni\ufb01ed auditory-tactile percept (the so-called \u201cHearing-Hands\nEffect\u201d).\n\nA simple extension to our model can reduce the number of connections below the requirements of\ntraditional dynamic routing models. Instead of having all-to-all connections between the input and\noutput layers, a relay layer of very few (M \u00bf N) neurons could be used to transmit the spikes\nform the input neurons to the output neurons (analogous to the single wire connecting encoder and\ndecoder in Figure 1B). A small number of (or ideally even one) relay neurons suf\ufb01ces because\nencoding and decoding occur in time. Hence, the connections between each input pool and the\nrelay neurons require O(M N) \u2248 O(N) connections (as long as M does not scale with N) and\nthose between the relay neurons and each output pool require O(M N) \u2248 O(N) connections as well.\nThus, by removing all-to-all connectivity between the input and output units (a standard feature in\ntraditional dynamic routing models), the number of required connections is reduced from O(N 2)\n\n7\n\n\fto O(N). Further, by replacing the strict pool boundaries with nearest neighbor connectivity in the\nprojection rings, the proposed model can accommodate a continuum of rotation angles.\n\nIn summary, we propose that the mechanism of dynamic routing through neuronal coherence could\nbe a general mechanism that could be used by multiple sensory and motor modalities in the neo-\ncortex: it is particularly suitable for placing raw information in an appropriate context (de\ufb01ned by\nthe routing tier).\n\nAcknowledgments\n\nDS was supported by a Stanford Graduate Fellowship and BP was supported under a National Science Foun-\ndation Graduate Research Fellowship.\n\nReferences\n\n[1] Olshausen B.A., Anderson C.H. & Van Essen D.C. (1993). A neurobiological model of visual attention and\ninvariant pattern recognition based on dynamic routing of information. Journal of Neuroscience 13(11):4700-\n4719.\n\n[2] Wiskott L. (2004). How does our visual system achieve shift and size invariance? 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Neuron 53(2):279-292.\n\n8\n\n\f", "award": [], "sourceid": 1109, "authors": [{"given_name": "Devarajan", "family_name": "Sridharan", "institution": null}, {"given_name": "Brian", "family_name": "Percival", "institution": null}, {"given_name": "John", "family_name": "Arthur", "institution": null}, {"given_name": "Kwabena", "family_name": "Boahen", "institution": null}]}