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Encoding certainty in bump attractors

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Persistent activity in neuronal populations has been shown to represent the spatial position of remembered stimuli. Networks that support bump attractors are often used to model such persistent activity. Such models usually exhibit translational symmetry. Thus activity bumps are neutrally stable, and perturbations in position do not decay away. We extend previous work on bump attractors by constructing model networks capable of encoding the certainty or salience of a stimulus stored in memory. Such networks support bumps that are not only neutrally stable to perturbations in position, but also perturbations in amplitude. Possible bump solutions then lie on a two-dimensional attractor, determined by a continuum of positions and amplitudes. Such an attractor requires precisely balancing the strength of recurrent synaptic connections. The amplitude of activity bumps represents certainty, and is determined by the initial input to the system. Moreover, bumps with larger amplitudes are more robust to noise, and over time provide a more faithful representation of the stored stimulus. In networks with separate excitatory and inhibitory populations, generating bumps with a continuum of possible amplitudes, requires tuning the strength of inhibition to precisely cancel background excitation.

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  • Amari, S. (1977) Dynamics of pattern formation in lateral-inhibition type neural fields. Biological Cybernetics, 27(2), 77–87.

    Article  CAS  PubMed  Google Scholar 

  • Armero, J., Casademunt J., Ramirez-Piscina, L., Sancho, J.M. (1998) Ballistic and diffusive corrections to front propagation in the presence of multiplicative noise. Physical Review E, 58, 5494–5500.

    Article  CAS  Google Scholar 

  • Basso, M.A., Wurtz, R.H. (1997) Modulation of neuronal activity by target uncertainty. Nature, 389(6646), 66–69.

    Article  CAS  PubMed  Google Scholar 

  • Beck, J.M., Ma, W.J., Kiani, R., Hanks, T., Churchland, A.K., Roitman, J., Shadlen, M.N., Latham, P.E., Pouget, A. (2008) Probabilistic population codes for bayesian decision making. Neuron, 60(6), 1142–1152.

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  • Ben-Yishai, R., Bar-Or, R.L., Sompolinsky, H. (1995) Theory of orientation tuning in visual cortex. Proceedings of the National Academy of Sciences of the United States of America, 92(9), 3844–3848.

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  • Bogacz, R., Brown, E., Moehlis, J., Holmes, P., Cohen, J.D. (2006) The physics of optimal decision making: a formal analysis of models of performance in two-alternative forced-choice tasks. Psychological Review, 113(4), 700–765.

    Article  PubMed  Google Scholar 

  • Bressloff, P.C. (2001) Traveling fronts and wave propagation failure in an inhomogeneous neural network. Physica D, 155(1–2), 83–100.

    Article  Google Scholar 

  • Bressloff, P.C. (2009) Stochastic neural field theory and the system-size expansion. SIAM Journal of Applied Mathematics, 70(5), 1488–1521.

    Article  Google Scholar 

  • Bressloff, P.C. (2012) Spatiotemporal dynamics of continuum neural fields. Journal of Physics A: Mathematical and Theoretical, 45(3), 033,001.

    Article  Google Scholar 

  • Bressloff, P.C., Webber, M.A. (2012) Front propagation in stochastic neural fields. SIAM Journal on Applied Dynamical Systems, 11(2), 708–740.

    Article  Google Scholar 

  • Brody, C.D., Romo, R., Kepecs, A. (2003) Basic mechanisms for graded persistent activity: discrete attractors, continuous attractors, and dynamic representations. Current Opinion in Neurobiology, 13(2), 204–211.

    Article  CAS  PubMed  Google Scholar 

  • Brunel, N., Wang, X.J. (2003) What determines the frequency of fast network oscillations with irregular neural discharges? I. synaptic dynamics and excitation-inhibition balance. Journal of Neurophysiology, 90(1), 415–430.

    Article  PubMed  Google Scholar 

  • Camperi, M., Wang, X.J. (1998) A model of visuospatial working memory in prefrontal cortex: recurrent network and cellular bistability. Journal of Computational Neuroscience, 5(4), 383–405.

    Article  CAS  PubMed  Google Scholar 

  • Churchland, A.K., Kiani, R., Shadlen, M.N. (2008) Decision-making with multiple alternatives. Nature Neuroscience, 11(6), 693–702.

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  • Compte, A., Brunel, N., Goldman-Rakic, P.S., Wang, X.J. (2000) Synaptic mechanisms and network dynamics underlying spatial working memory in a cortical network model. Cerebral Cortex, 10(9), 910–923.

    Article  CAS  PubMed  Google Scholar 

  • Constantinidis, C., Wang, X.J. (2004) A neural circuit basis for spatial working memory. Neuroscientist, 10(6), 553–565.

    Article  PubMed  Google Scholar 

  • Coombes, S., Owen, M.R. (2004) Evans functions for integral neural field equations with heaviside firing rate function. SIAM Journal on Applied Dynamical Systems, 3(4), 574–600.

    Article  Google Scholar 

  • Curtis, C.E. (2006) Prefrontal and parietal contributions to spatial working memory. Neuroscience, 139(1), 173–180.

    Article  CAS  PubMed  Google Scholar 

  • Durstewitz, D., Seamans, J.K., Sejnowski, T.J. (2000) Neurocomputational models of working memory. Nature Neuroscience, 3 Supplement, 1184–1191.

    Article  Google Scholar 

  • El Boustani, S., Destexhe, A. (2009) A master equation formalism for macroscopic modeling of asynchronous irregular activity states. Neural Computation, 21(1), 46–100.

    Article  PubMed  Google Scholar 

  • Ermentrout, B. (1998) Neural networks as spatio-temporal pattern-forming systems. Reports on Progress in Physics, 61(4), 353.

    Article  Google Scholar 

  • Faisal, A., Selen, L., Wolpert, D. (2008) Noise in the nervous system. Nature Reviews Neuroscience, 9(4), 292–303.

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  • Folias, S.E., Bressloff, P.C. (2004) Breathing pulses in an excitatory neural network. SIAM Journal on Applied Dynamical Systems, 3(3), 378–407.

    Article  Google Scholar 

  • Funahashi, S., Bruce, C.J., Goldman-Rakic, P.S. (1989) Mnemonic coding of visual space in the monkey’s dorsolateral prefrontal cortex. Journal of Neurophysiology, 61(2), 331–349.

    CAS  PubMed  Google Scholar 

  • Ginzburg, I., Sompolinsky. (1994) Theory of correlations in stochastic neural networks. Physical Review E, 50(4), 3171–3191.

    Article  CAS  Google Scholar 

  • Gold, J.I., Shadlen, M.N. (2002) Banburismus and the brain: decoding the relationship between sensory stimuli, decisions, and reward. Neuron, 36(2), 299–308.

    Article  CAS  PubMed  Google Scholar 

  • Goldman, M.S., Levine, J.H., Major, G., Tank, D.W., Seung, H.S. (2003) Robust persistent neural activity in a model integrator with multiple hysteretic dendrites per neuron. Cerebral Cortex, 13(11), 1185–1195.

    Article  PubMed  Google Scholar 

  • Goldman-Rakic, P.S. (1995) Cellular basis of working memory. Neuron, 14(3), 477–485.

    Article  CAS  PubMed  Google Scholar 

  • Gutkin, B., Laing, C., Colby, C., Chow, C., Ermentrout, G. (2001) Turning on and off with excitation: The role of spike-timing asynchrony and synchrony in sustained neural activity. Journal of Computational Neuroscience, 11(2), 121–134.

    Article  CAS  PubMed  Google Scholar 

  • Haider, B., Duque, A., Hasenstaub, A.R., McCormick, D.A. (2006) Neocortical network activity in vivo is generated through a dynamic balance of excitation and inhibition. Journal of Neuroscience, 26(17), 4535–4545.

    Article  CAS  PubMed  Google Scholar 

  • Hansel, D., Mato, G. (2013) Short-term plasticity explains irregular persistent activity in working memory tasks. Journal of Neuroscience, 33(1), 133–149.

    Article  CAS  PubMed  Google Scholar 

  • Hansel, D., Sompolinsky, H. (1998) Modeling feature selectivity in local cortical circuits. In C. Koch, & I. Segev (Eds.) Methods in neuronal modeling: From ions to networks (pp. 499–567). Cambridge: MIT, chap 13.

    Google Scholar 

  • Hansel, D., van Vreeswijk, C. (2012) The mechanism of orientation selectivity in primary visual cortex without a functional map. Journal of Neuroscience, 32(12), 4049–4064. doi: 10.1523/JNEUROSCI.6284-11.2012.

    Article  CAS  PubMed  Google Scholar 

  • Hutt, A., Longtin, A., Schimansky-Geier, L. (2008) Additive noise-induced turing transitions in spatial systems with application to neural fields and the swift–hohenberg equation. Physica D, 237(6), 755–773.

    Article  Google Scholar 

  • Itskov, V., Hansel, D., Tsodyks, M. (2011) Short-term facilitation may stabilize parametric working memory trace. Frontiers in Computational Neuroscience, 5, 40.

    Article  PubMed Central  PubMed  Google Scholar 

  • Kiani, R., Shadlen, M.N. (2009) Representation of confidence associated with a decision by neurons in the parietal cortex. Science, 324(5928), 759–764.

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  • Kilpatrick, Z.P., Bressloff, P.C. (2010) Effects of adaptation and synaptic depression on spatiotemporal dynamics of an excitatory neuronal network. Physica D, 239, 547–560.

    Article  Google Scholar 

  • Kilpatrick, Z.P., Ermentrout, B, (2013) Wandering bumps in stochastic neural fields. SIAM Journal on Applied Dynamical Systems, 12, 61–94.

    Article  Google Scholar 

  • Kilpatrick, Z.P., Ermentrout, B. Doiron, B. (2013) Optimizing working memory with heterogeneity of recurrent cortical excitation. Journal of Neuroscience, in press.

  • Laing, C.R., Chow, C.C. (2001) Stationary bumps in networks of spiking neurons. Neural Computation, 13(7), 1473–1494.

    Article  CAS  PubMed  Google Scholar 

  • Machens, C., Romo, R., Brody, C. (2005) Flexible control of mutual inhibition: a neural model of two-interval discrimination. Science, 307(5712), 1121–1124.

    Article  CAS  PubMed  Google Scholar 

  • Meyer, T., Qi, X.L., Stanford, T.R., Constantinidis, C. (2011) Stimulus selectivity in dorsal and ventral prefrontal cortex after training in working memory tasks. Journal of Neuroscience, 31(17), 6266–6276.

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  • Miller, E.K., Erickson, C.A., Desimone, R. (1996) Neural mechanisms of visual working memory in prefrontal cortex of the macaque. Journal of Neuroscience, 16(16), 5154–5167.

    CAS  PubMed  Google Scholar 

  • Pesaran, B., Pezaris, J.S., Sahani, M., Mitra, P.P., Andersen, R.A. (2002) Temporal structure in neuronal activity during working memory in macaque parietal cortex. Nature Neuroscience, 5(8), 805–811.

    Article  CAS  PubMed  Google Scholar 

  • Pinto, D.J., Ermentrout, G.B. (2001a) Spatially structured activity in synaptically coupled neuronal networks: I. Traveling fronts and pulses. SIAM Journal of Applied Mathematics, 62(1), 206–225.

    Article  Google Scholar 

  • Pinto, D.J., Ermentrout, G.B. (2001b) Spatially structured activity in synaptically coupled neuronal networks: Ii. lateral inhibition and standing pulses. SIAM Journal of Applied Mathematics, 62(1), 226–243.

    Article  Google Scholar 

  • Ploner, C.J., Gaymard, B., Rivaud, S., Agid, Y., Pierrot-Deseilligny, C. (1998) Temporal limits of spatial working memory in humans. European Journal Neuroscience, 10(2), 794–797.

    Article  CAS  Google Scholar 

  • Polk, A., Litwin-Kumar, A., Doiron, B. (2012) Correlated neural variability in persistent state networks. Proceedings of the National Academy of Sciences of the United States of America, 109(16), 6295–6300.

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  • Rao, S.G., Williams, G.V., Goldman-Rakic, P.S. (2000) Destruction and creation of spatial tuning by disinhibition: GABA(A) blockade of prefrontal cortical neurons engaged by working memory. Journal of Neuroscience, 20(1), 485–494.

    CAS  PubMed  Google Scholar 

  • Renart, A ., Song, P., Wang, X.J. (2003) Robust spatial working memory through homeostatic synaptic scaling in heterogeneous cortical networks. Neuron, 38(3), 473–485.

    Article  CAS  PubMed  Google Scholar 

  • Renart, A., de la Rocha, J., Bartho, P., Hollender, L., Parga, N., Reyes, A., Harris, K.D. (2010) The asynchronous state in cortical circuits. Science, 327(5965), 587–590. doi:10.1126/science.1179850.

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  • Sandstede, B. (2002) Stability of travelling waves. Handbook of dynamical systems, 2, 983–1055.

    Article  Google Scholar 

  • Seung, H.S. (1996) How the brain keeps the eyes still. Proceedings of the National Academy of Sciences of the United States of America, 13(23), 13,339–13,344.

    Article  Google Scholar 

  • Shadlen, M.N., Newsome, W.T. (1998) The variable discharge of cortical neurons: implications for connectivity, computation, and information coding. Journal of Neuroscience, 18(10), 3870– 3896.

    CAS  PubMed  Google Scholar 

  • Shew, W.L., Yang, H., Petermann, T., Roy, R., Plenz, D. (2009) Neuronal avalanches imply maximum dynamic range in cortical networks at criticality. Journal of Neuroscience, 15(49), 595–600.

    Google Scholar 

  • Softky, W.R., Koch, C. (1993) The highly irregular firing of cortical cells is inconsistent with temporal integration of random epsps. Journal of Neuroscience, 13(1), 334–350.

    CAS  PubMed  Google Scholar 

  • Somogyi, P., Tamás, G., Lujan, R., Buhl, E.H. (1998) Salient features of synaptic organisation in the cerebral cortex. Brain Research. Brain Research Reviews, 26(2–3), 113–135.

    Article  CAS  PubMed  Google Scholar 

  • Veltz, R., Faugeras, O. (2010) Local/global analysis of the stationary solutions of some neural field equations. SIAM Journal on Applied Dynamical Systems, 9, 954–998.

    Article  Google Scholar 

  • Vijayraghavan, S., Wang, M., Birnbaum, S.G., Williams, G.V., Arnsten, A.F.T. (2007) Inverted-u dopamine d1 receptor actions on prefrontal neurons engaged in working memory. Nature Neuroscience, 10(3), 376–384.

    Article  CAS  PubMed  Google Scholar 

  • van Vreeswijk, C., Sompolinsky, H. (1996) Chaos in neuronal networks with balanced excitatory and inhibitory activity. Science, 274(5293), 1724–1726.

    Article  PubMed  Google Scholar 

  • Wang, X.J. (2001) Synaptic reverberation underlying mnemonic persistent activity. Trends in Neurosciences, 24(8), 455–463.

    Article  CAS  PubMed  Google Scholar 

  • Wang, X.J. (2002) Probabilistic decision making by slow reverberation in cortical circuits. Neuron, 36(5), 955–968.

    Article  CAS  PubMed  Google Scholar 

  • White, J.A., Rubinstein, J.T., Kay, A.R. (2000) Channel noise in neurons. Trends in Neurosciences, 23(3), 131–137.

    Article  CAS  PubMed  Google Scholar 

  • White, J.M., Sparks, D.L., Stanford, T.R. (1994) Saccades to remembered target locations: an analysis of systematic and variable errors. Vision Research, 34(1), 79–92.

    Article  CAS  PubMed  Google Scholar 

  • Wilson, H.R., Cowan, J.D. (1973) A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue. Biological Cybernetics, 13(2), 55–80.

    CAS  Google Scholar 

  • Yizhar, O., Fenno, L.E., Prigge, M., Schneider, F., Davidson, T.J., O’Shea, D.J., Sohal, V.S., Goshen, I., Finkelstein, J., Paz, J.T., Stehfest, K., Fudim, R., Ramakrishnan, C., Huguenard, J.R., Hegemann, P., Deisseroth, K. (2011) Neocortical excitation/inhibition balance in information processing and social dysfunction. Nature, 477(7363), 171–178.

    Article  CAS  PubMed  Google Scholar 

  • Zemel, R.S., Dayan, P., Pouget, A. (1998) Probabilistic interpretation of population codes. Neural Computation, 10(2), 403–430.

    Article  CAS  PubMed  Google Scholar 

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Correspondence to Zachary P. Kilpatrick.

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Appendix A: Stability analysis of stationary bumps

The stability of a stationary bump solution U(x) is determined by writing

$$ u(x,t) = U(x) + \bar{\psi}(x,t),$$

where \(||\bar {\psi }(x,t)|| \ll 1\) and expanding Eq. (2.1) to first-order in \(\bar {\psi }(x,t)\). This leads to the linear equation

$$\begin{array}{@{}rcl@{}} \frac{\partial \bar{\psi}(x,t)}{\partial t} = - \bar{\psi}(x,t) + \int_{- \pi}^{\pi} w(x-y) f'(U(y)) \bar{\psi}(y,t) \mathrm{d} y.\\ \end{array}$$

Since Eq. (A.2) is linear, we can use separation of variables to characterize all of its solutions (Ermentrout 1998; Sandstede 2002; Folias and Bressloff 2004; Veltz and Faugeras 2010; Bressloff 2012). Plugging the ansatz \(\bar {\psi }(x,t) = b(t) \psi (x)\) into Eq. (A.2), we find

$$\begin{array}{@{}rcl@{}} b'(t) \psi (x) = - b(t) \psi(x,t){\kern50pt}\\ + b(t) \int_{- \pi}^{\pi} w(x-y) f'(U(y)) \psi(y) \mathrm{d} y.{}\end{array}$$

After rearranging terms, we have

$$\begin{array}{@{}rcl@{}} \frac{b'(t)}{b(t)} = - 1 + \frac{1}{\psi(x)} \int_{- \pi}^{\pi} w(x-y) f'(U(y)) \psi(y) \mathrm{d} y, \end{array}$$

meaning that both the left and right hand sides of Eq. (A.3) equal a constant λ. Therefore b′(t) = λb(t) implying b(t) = e λt, and

$$\begin{array}{@{}rcl@{}} (\lambda + 1) \psi (x) = \int_{- \pi}^{\pi} w(x-y) f'(U(y)) \psi(y) \mathrm{d} y, \end{array}$$

which is an eigenvalue problem characterizing the stability of bump solutions to Eq. (2.1). Since U(x) < 0 over some portion of the domain, the function f′(U(x)) will have jump discontinuities, but since the domain − π, π is periodic, we can always rearrange the limits of integration to ensure these jump discontinuity points are on the interior of the integral.To characterize the eigensolutions to Eq. (A.4), we decompose the function ψ(x) into a Fourier series on the domain − π, π given by

$$\begin{array}{@{}rcl@{}} \psi(x) = \sum\limits_{k=1}^{N}\mathcal{A}_{k}\cos{kx}+\sum\limits_{k=1}^{N}\mathcal{B}_{k}\sin{kx}, \end{array}$$

where N is directly determined by the number of terms in the Fourier expansion of w(x). The associated coefficients of the expansion in Eq. (A.5) are then determined by the linear system

$$\begin{array}{@{}rcl@{}} \mathcal{A}_{k} &=& w_{k} \int_{- \pi}^{\pi} \cos (kx) f'(U(x)) \psi (x) \mathrm{d} x,\\ \mathcal{B}_{l} &=& w_{l} \int_{- \pi}^{\pi} \sin (lx) f'(U(x)) \psi (x) \mathrm{d} x, \end{array}$$

where k, l = 1, . . . , N. Solutions of this system, along with the associated λ are eigensolutions of Eq. (A.4). We can directly compute the eigenvalues associated with the stability of bumps in the case of the weight function in Eq. (2.2) so that

$$\begin{array}{@{}rcl@{}} (\lambda+1)\psi(x) = w_{1}\int_{-\pi}^{\pi}\cos{(x-y)}f'(U(y))\psi(y)dy. \\ \end{array}$$

Analyzing solutions (λ, ψ) of Eq. (A.6) is equivalent to determining the elements of the spectrum of the linear system in the vicinity of the bump. We are mainly interested in the point spectrum of the linear operator in Eq. (A.6), since the sign of the real part of λ for these solutions will determine the associated stability of stationary bump solutions (see Coombes and Owen 2004; Veltz and Faugeras 2010) for detailed discussions of the partitioning of spectra in neural field models. In particular, we examine the stability of stationary bump solutions of the form U(x) = A cosxwhen the firing rate function has the form given in Eq. (2.3). Hence,

$$\begin{array}{@{}rcl@{}} (\lambda+1)\mathcal{A}_{1} &=& \left\{\begin{array}{ll} \mathcal{A}_1sw_{1}\left[\frac{\pi}{2}-\cos^{-1}\left(\frac{1}{sA}\right)-\frac{1}{sA}\sqrt{1-\left(\frac{1}{sA}\right)^{2}}\right], & \indent \text{for}\;\; sA > 1, \\ \mathcal{A}_1sw_{1}\frac{\pi}{2}, & \indent \text{for}\;\; sA \le 1, \end{array}\right. \end{array}$$
$$\begin{array}{@{}rcl@{}} (\lambda+1)\mathcal{B}_{1} & = & \left\{\begin{array}{ll} \mathcal{B}_1sw_{1}\left[\frac{\pi}{2}-\cos^{-1}\left(\frac{1}{sA}\right)+\frac{1}{sA}\sqrt{1-\left(\frac{1}{sA}\right)^{2}}\right], & \indent \text{for}\;\; sA > 1 ,\\ \mathcal{B}_1sw_{1}\frac{\pi}{2}, & \indent \text{for}\;\; sA \le 1, \end{array}\right. \end{array}$$

and \(\mathcal {A}_{k} = \mathcal {B}_{k} = 0\) for k ≠ 1 and \(\mathcal {A}_{0} = 0\). Therefore, only sinx and cosx are eigenfunctions of the linearized system. All other Fourier modes cos(kx) and sin(lx) are linear combinations of functions associated with the essential spectrum λ = − 1, given by \(\psi (x) = \cos (kx) - \mathcal {C}_{k} \cos (x)\) and \(\psi (x) = \sin (lx) - \mathcal {D}_{k} \sin (x)\) where

$$\begin{array}{@{}rcl@{}}</p><p class="noindent">\mathcal{C}_k&=&\frac{\displaystyle\int_{- \pi}^{\pi} \cos y f'(U(y)) \cos (ky) \d y}{\displaystyle \int_{- \pi}^{\pi} \cos^{2} y f'(U(y)) \mathrm{d} y},\\ \mathcal{D}_k&=&\frac{\displaystyle\int_{- \pi}^{\pi} \sin y f'(U(y)) \sin (ky) \mathrm{d} y}{\displaystyle\int_{- \pi}^{\pi} \sin^{2} y f'(U(y)) \mathrm{d} y} \end{array}$$

as well as the eigenfunctions cos(x) and sin(x). Now, bump solutions of Eq. (2.1) will be neutrally stable to both even and odd perturbations when parameters in Eqs. (A.7A.8) are such that some solutions have Reλ = 0 and others have Reλ < 0. When A > 1/s, we find

$$\begin{array}{@{}rcl@{}} \lambda_{o} &=& \frac{2}{\pi}\left(-\cos^{-1}\left(\frac{1}{sA}\right)-\frac{1}{sA}\sqrt{1-\left(\frac{1}{sA}\right)^{2}}\right) < 0, \\ \lambda_{e} & = & \frac{2}{\pi}\left(-\cos^{-1}\left(\frac{1}{sA}\right)+\frac{1}{sA}\sqrt{1-\left(\frac{1}{sA}\right)^{2}}\right) < 0. \end{array}$$

Appendix B: Existence and stability of bumps in two-population network

To find stationary bump solutions to the excitatory-inhibitory network defined by Eq. (2.4) with synaptic weights given by Eq. (2.5), we make the ansatz

$$\begin{array}{@{}rcl@{}} u(x,t) =& U(x) = A_{0} + A_{1}\cos{x}, \\ v(x,t) =& V(x) = M_{0} + M_{1}\cos{x}. \end{array}$$

and substitute the v equation into the u equation in Eq. (2.4) to generate

$$\begin{array}{@{}rcl@{}} U(x) = (w_{ee}(x)-w_{ei}(x)*w_{ie}(x))*f(U(x)), \end{array}$$

where \(f(x)*g(x) = \int _{-\pi }^{\pi }f(x-y)g(y)dy\). Therefore, stationary solutions to Eq. (2.4) are the same as stationary solutions to Eq. (2.1) by assigning

$$\begin{array}{@{}rcl@{}} w(x) &=& w_{ee}(x)-w_{ei}(x)*w_{ie}(x)\\ &=& \bar{w}_{ee}-2\pi\bar{w}_{ei}\bar{w}_{ie}+\bar{w}_{ee}\cos{x}.\\ \end{array}$$

Note that Eq. (B.3) is equivalent to Eq. (2.2) by setting \(w_{0} = \bar {w}_{ee}-2\pi \bar {w}_{ei}\bar {w}_{ie}\) and \(w_{1} = \bar {w}_{ee}\). Therefore, under an appropriate change of variables, solving Eq. (B.2) is equivalent to solving Eq. (3.1). Therefore, our results concerning the existence of a continuum of amplitudes concerning Eq. (3.1) should hold here as well. This means that in order to obtain a line attractor of bump amplitudes, we must have that A 0 = 0 and \(\bar {w}_{ee}=2\pi \bar {w}_{ei}\bar {w}_{ie}\) (i.e., w 0 = 0). However, we can still have M 0 ≠ 0. Additionally, analogous to the single network in Eq. (3.1), we must require that \(\bar {w}_{ee} = \frac {2}{\pi s}\). Again, we have

$$\begin{array}{@{}rcl@{}} A =\left\{\begin{array}{ll} sA\bar{w}_{ee}\left[\frac{\pi}{2}-\cos^{-1}\left(\frac{1}{sA}\right)-\frac{1}{sA}\sqrt{1-\left(\frac{1}{sA}\right)^{2}}\right]+2\bar{w}_{ee}\sqrt{1-\left(\frac{1}{sA}\right)^{2}}, & \indent \text{for}\;\; sA > 1, \\ sA\bar{w}_{ee}\frac{\pi}{2}, & \indent \text{for}\;\; sA \le 1, \end{array}\right. \end{array}$$

and, for the v equation

$$\begin{array}{@{}rcl@{}}</p><p class="noindent">V(x) = \bar{w}_{ei}\int_{-\pi}^{\pi}(1+\cos{(x-y)})f(U(y))dy, \end{array}$$

so that

$$\begin{array}{@{}rcl@{}} M_{0} &=& \left\{\begin{array}{ll} 2s\bar{w}_{ei} A\left[1-\sqrt{1-\left(\frac{1}{sA}\right)^{2}}\right] \\{\kern3pc} +2\bar{w}_{ei}\cos^{-1}\left(\frac{1}{sA}\right), & {\kern3.5pc} \text{for}\;\; sA > 1, \\ 2s\bar{w}_{ei} A, & {\kern3.5pc} \text{for}\;\; sA \le 1, \end{array}\right. \\ M_{1} & = & \left\{\begin{array}{ll} sA\bar{w}_{ei}\left[\frac{\pi}{2}-\cos^{-1}\left(\frac{1}{sA}\right)-\frac{1}{sA}\sqrt{1-\left(\frac{1}{sA}\right)^{2}}\right]\\{\kern3pc}+2\bar{w}_{ei}\sqrt{1-\left(\frac{1}{sA}\right)^{2}}, &{} \text{for}\;\; sA > 1, \\ sA\bar{w}_{ei}\frac{\pi}{2}, &{} \text{for}\;\; sA \le 1. \\ \end{array}\right.\\ \end{array}$$

Again, we have a continuum of values for \(A \in [0, \frac {\pi \bar {w}_{ee}}{2}]\) that are fixed points, and the coefficients for v will depend on A, and upon substituting values for s we obtain

$$M_{0} = \frac{4\bar{w}_{ie}}{\pi\bar{w}_{ee}} A, \text{\indent} M_{1} = \frac{\bar{w}_{ie}}{\bar{w}_{ee}}A.$$

To study the way in which the line attractor globally organizes dynamics, we consider effects of breaking this balance condition in two ways: excess excitation or excess inhibition. As we shall see, too much inhibition leads to no stable bump solutions whereas too much excitation leads to only a single stable bump solution. To do this, we define the quantity \(\bar {w} = \bar {w}_{ee} - 2\pi \bar {w}_{ei}\bar {w}_{ie}\) and simply consider when > 0 (excess excitation) and < 0 (excess inhibition).

First let < 0 (excess inhibition) and consider when U(x) < 1/s. Then, similar to Section 3.1, we find that

$$\begin{array}{@{}rcl@{}}A_{0} & = & 2s\bar{w}[aA_0+\sin{a}A_{1}] \\ A_{1} & = & s\bar{w}_{ee}[\sin{a}A_0+aA_{1}] \end{array}$$

where \(a = \cos ^{-1}\left (-\frac {A_{0}}{A_{1}}\right )\) and |A 0| ≤ |A 1| . We must consider the cases when A 0 > 0, A 0 < 0 and A 0 = 0. If A 0 > 0, then since 0 ≤ aπ we know that sina ≥ 0. Also, we impose that A 1 ≥ 0 so that the peak of the bump corresponds to the remembered location of the stimulus. Then, since < 0, Eq. (B.6) implies that A 0 equals something negative, which is a contradiction. Now assume that A 0 < 0. Then Eq. (B.6) implies that

$$a \ge \frac{A_{1}}{|A_0|}\sin{a}, \qquad \text{and} \qquad a \ge \frac{|A_0|}{A_{1}}\sin{a},$$

which implies that |A 0| = A 1. Then our only choices are U(x) = A 1(cosx − 1) or U(x) ≡ 0. However, if the former were true, then f(u) ≡ 0 which forces U(x) ≡ 0. Finally it is easy to see that if A 0 = 0 then A 1 = 0 for ≠ 0.

Now assume that > 0 (excess excitation). In the case U(x) < 1/s, we find that the only solution is U(x) ≡ 0. When U(x) > 1/s for some x, then Eq. (B.6) becomes

$$\begin{array}{@{}rcl@{}} A_{0} & = & 2s\bar{w}[(a-b)A_{0} + (\sin{a}-\sin{b})A_{1}] + 2\bar{w}b, \\ A_{1} & = & s\bar{w}_{ee}[(\sin{a}-\sin{b})A_{0} + (a-b)A_{1}] + 2\bar{w}_{ee}\sin{b},\\ \end{array}$$

where \(b = \cos ^{-1}\left (\frac {1-sA_{0}}{sA_{1}}\right )\) such that U(b) = 1/s. To simplify the analysis, we will let \(s = \frac {2}{\pi \bar {w}_{ee}}\) as was the condition for the line attractor.

We now perform a stability analysis on the fixed bump solution in Eq. (B.1). We consider the set of parameters A 0 = 0 and A 1 = A ∈ [0, 1/s] that leads to a line attractor of amplitudes. Similar to Section 3.2, we study the temporal evolution of perturbations to the original bump solutions by plugging in the linear expansion

$$\begin{array}{@{}rcl@{}} u(x,t) & = & U(x) + \bar{\psi}(x,t), \\ v(x,t) & = & V(x) + \bar{\phi}(x,t), \end{array}$$

where \(||\bar {\psi }(x,t)||,||\bar {\phi }(x,t)|| \ll 1\). As before, we can show that these solutions are separable, so that \(\bar {\psi }(x,t) = \mathrm {e}^{\lambda t} \psi (x) \) and \(\bar {\phi }(x,t) = \mathrm {e}^{\lambda t} \phi (x)\). Therefore, by plugging the expansion Eq. (B.8) into Eq. (2.4), and noting separability, we find

$$\begin{array}{@{}rcl@{}} (\lambda+1)\psi(x) & = & w_{ee}{\ast}(f'(U(x))\psi(x))- w_{ie}{\ast}\phi(x), \\ (\tau\lambda+1)\phi(x) & = & w_{ei}{\ast}(f'(U(x))\psi(x)). \end{array}$$

We then expand both spatial functions in Fourier series

$$\begin{array}{@{}rcl@{}} \psi(x) & = & \sum\limits_{k=0}^{N}\mathcal{A}_{k}\cos{(kx)}+\sum\limits_{k=1}^{N}\mathcal{B}_{k}\sin{(kx)}, \\ \phi(x) & = & \sum\limits_{k=0}^{N}\mathcal{M}_{k}\cos{(kx)}+\sum\limits_{k=1}^{N}\mathcal{N}_{k}\sin{(kx)}. \end{array}$$

Similar to Section 3, we analyze the solutions (λ, ψ, ϕ) to determine the stability of the perturbations by observing the sign of the real part of λ. By plugging Eq. (B.10)intoEq. (B.9), we see that when using the weight functions in Eq. (2.5) we have the system

$$\begin{array}{@{}rcl@{}} (\lambda + 1)\mathcal{A}_{0} & = & \bar{w}_{ee}\int_{-\pi}^{\pi}(\mathcal{A}_{0} + \mathcal{A}_{1}\cos{y}+\mathcal{B}_{1}\sin{y})f'(U(y))dy\\ &&-\bar{w}_{ie}\int_{-\pi}^{\pi}(\mathcal{M}_0+\mathcal{M}_{1}\cos{y}+\mathcal{N}_{1}\sin{y})dy, \\ (\lambda+1)\mathcal{A}_{1} & = & \bar{w}_{ee}\int_{-\pi}^{\pi}\cos{y}(\mathcal{A}_{0} + \mathcal{A}_{1}\cos{y}+\mathcal{B}_{1}\sin{y})f'(U(y))dy, \\ (\lambda+1)\mathcal{B}_{1} & = & \bar{w}_{ee}\int_{-\pi}^{\pi}\sin{y}(\mathcal{A}_{0} + \mathcal{A}_{1}\cos{y}+\mathcal{B}_{1}\sin{y})f'(U(y))dy, \\ (\tau\lambda+1)\mathcal{M}_{0} & = & \bar{w}_{ei}\int_{-\pi}^{\pi}(\mathcal{A}_{0} + \mathcal{A}_{1}\cos{y}+\mathcal{B}_{1}\sin{y})f'(U(y))dy, \\ (\tau\lambda+1)\mathcal{M}_{1} & = & \bar{w}_{ei}\int_{-\pi}^{\pi}\cos{y}(\mathcal{A}_{0} + \mathcal{A}_{1}\cos{y}+\mathcal{B}_{1}\sin{y})f'(U(y))dy, \\ (\tau\lambda+1)\mathcal{N}_{1} & = & \bar{w}_{ei}\int_{-\pi}^{\pi}\sin{y}(\mathcal{A}_{0} + \mathcal{A}_{1}\cos{y}+\mathcal{B}_{1}\sin{y})f'(U(y))dy,\\ \end{array}$$

where \(\mathcal {A}_{k} = \mathcal {B}_{k} = 0\) for k ≠ 0, 1. When τ ≠ 0, we can compute the integrals and set conditions of the parameters for the line attractor to find that the system in Eq. (B.11) is equivalent to the linear system

$$\begin{array}{@{}rcl@{}} \lambda \left(\begin{array}{l} \mathcal{A}_{0} \\ \mathcal{A}_{1} \\ \mathcal{B}_{1} \\ \mathcal{M}_{0} \\ \mathcal{M}_{1} \\ \mathcal{N}_1 \end{array}\right) = \left(\begin{array}{cccccc} 1 & \frac{4}{\pi} & 0 & -2\pi\bar{w}_{ie} & 0 & 0 \\ \frac{4}{\pi} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \frac{\displaystyle 1}{\displaystyle \pi\bar{w}_{ie}\tau} & \frac{\displaystyle 2}{\displaystyle \pi^{2}\bar{w}_{ie}\tau} & 0 & -\frac{\displaystyle 1}{\displaystyle \tau} & 0 & 0 \\ \frac{\displaystyle 2}{\displaystyle \pi^{2}\bar{w}_{ie}\tau} & \frac{\displaystyle 1}{\displaystyle 2\pi\bar{w}_{ie}\tau} & 0 & 0 & -\frac{\displaystyle 1}{\displaystyle \tau} & 0 \\ 0 & 0 & \frac{\displaystyle 1}{\displaystyle 2\pi\bar{w}_{ie}\tau} & 0 & 0 & -\frac{\displaystyle 1}{\displaystyle \tau} \end{array}\right) \left(\begin{array}{l} \mathcal{A}_{0} \\ \mathcal{A}_{1} \\ \mathcal{B}_{1} \\ \mathcal{M}_{0} \\ \mathcal{M}_{1} \\ \mathcal{N}_{1} \end{array}\right). \end{array}$$

The associated matrix has the characteristic equation


from which we obtain only two zero eigenvalues corresponding to odd perturbations \(\left (0,0,1,0,0,\frac {\bar {w}_{ei}}{\bar {w}_{ee}}\right )\) and even perturbations \(\left (0,1,0,\frac {4\bar {w}_{ei}}{\pi \bar {w}_{ee}},\frac {\bar {w}_{ei}}{\bar {w}_{ee}},0\right )\). Thus we see that obtaining a zero eigenvalue associated with even perturbations does not depend on the speed of inhibition, τ. However, neutral stability still does depend on τ, as it is possible that other eigenvalues associated with even perturbations may have positive real part. Looking at the other eigenvalues, we have two negative ones, \(\lambda _{-} = -\frac {1}{\tau }\), corresponding to perturbations in \(\mathcal {M}_{1}\) and \(\mathcal {N}_{1}\). Therefore, if we only perturb the inhibitory network, then solutions will be attracted back toward the fixed bump solutions. The final two eigenvalues can be analyzed by examining

$$\begin{array}{@{}rcl@{}} \lambda_{\pm} = \frac{1}{2}\left(1-\frac{1}{\tau}\right) \pm \frac{1}{2\tau}\sqrt{\left(1+\frac{64}{\pi^{2}}\right)\tau^{2} - 6\tau + 1}. \end{array}$$

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Carroll, S., Josić, K. & Kilpatrick, Z.P. Encoding certainty in bump attractors. J Comput Neurosci 37, 29–48 (2014).

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