Journal of Computational Neuroscience

, Volume 46, Issue 2, pp 145–168 | Cite as

Network structure and input integration in competing firing rate models for decision-making

  • Victor J. BarrancaEmail author
  • Han Huang
  • Genji Kawakita


Making a decision among numerous alternatives is a pervasive and central undertaking encountered by mammals in natural settings. While decision making for two-option tasks has been studied extensively both experimentally and theoretically, characterizing decision making in the face of a large set of alternatives remains challenging. We explore this issue by formulating a scalable mechanistic network model for decision making and analyzing the dynamics evoked given various potential network structures. In the case of a fully-connected network, we provide an analytical characterization of the model fixed points and their stability with respect to winner-take-all behavior for fair tasks. We compare several means of input integration, demonstrating a more gradual sigmoidal transfer function is likely evolutionarily advantageous relative to binary gain commonly utilized in engineered systems. We show via asymptotic analysis and numerical simulation that sigmoidal transfer functions with smaller steepness yield faster response times but depreciation in accuracy. However, in the presence of noise or degradation of connections, a sigmoidal transfer function garners significantly more robust and accurate decision-making dynamics. For fair tasks and sigmoidal gain, our model network also exhibits a stable parameter regime that produces high accuracy and persists across tasks with diverse numbers of alternatives and difficulties, satisfying physiological energetic constraints. In the case of more sparse and structured network topologies, including random, regular, and small-world connectivity, we show the high-accuracy parameter regime persists for biologically realistic connection densities. Our work shows how neural system architecture is potentially optimal in making economic, reliable, and advantageous decisions across tasks.


Network structure Firing rate models Nonlinear dynamics Decision-Making Input integration 



This work was supported by NSF grant DMS-1812478 and a Swarthmore Faculty Research Support Grant.

Compliance with Ethical Standards

Conflict of interests

The authors declare that they have no conflict of interest.


  1. Ahmed, B., Anderson, J.C., Douglas, R.J., Martin, K.A., Whitteridge, D. (1998). Estimates of the net excitatory currents evoked by visual stimulation of identified neurons in cat visual cortex. Cereb Cortex, 8(5), 462–476.PubMedGoogle Scholar
  2. Andronov, A.A. (1973). Qualitative theory of second-order dynamic systems. A Halsted Press book. Wiley. ISBN 9780706512922.Google Scholar
  3. Barranca, V.J., Zhou, D., Cai, D. (2015a). A novel characterization of amalgamated networks in natural systems. Scientific Reports, 5, 10611.Google Scholar
  4. Barranca, V.J., Zhou, D., Cai, D. (2015b). Low-rank network decomposition reveals structural characteristics of small-world networks. Phys rev e stat nonlin soft matter phys, 92(6), 062822.Google Scholar
  5. Barttfeld, P., Uhrig, L., Sitt, J.D., Sigman, M., Jarraya, B., Dehaene, S. (2015). Signature of consciousness in the dynamics of resting-state brain activity. Proceedings of the National Academy of Sciences of the USA, 112(3), 887–892. ISSN 0027-8424. Scholar
  6. Bendixson, I. (1901). Sur les courbes definies par des equations differentielles. Acta Math, 24, 1–88. Scholar
  7. Binas, J., Rutishauser, U., Indiveri, G., Pfeiffer, M. (2014). Learning and stabilization of winner-take-all dynamics through interacting excitatory and inhibitory plasticity. Frontiers in Computational Neuroscience, 8, 68.PubMedPubMedCentralGoogle Scholar
  8. Bogacz, R., Usher, M., Zhang, J., McClelland, J.L. (2007). Extending a biologically inspired model of choice: multi-alternatives, nonlinearity and value-based multidimensional choice. Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences, 362(1485), 1655–1670.PubMedPubMedCentralGoogle Scholar
  9. Bogacz, R., Wagenmakers, E.J., Forstmann, B.U., Nieuwenhuis, S. (2010). The neural basis of the speed-accuracy tradeoff. Trends in Neurosciences, 33(1), 10–16.PubMedGoogle Scholar
  10. 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.PubMedGoogle Scholar
  11. Brouwer, L.E.J. (1912). Über abbildung von mannigfaltigkeiten. Mathematische Annalen, 71(4), 598–598. ISSN 1432-1807. Scholar
  12. Churchland, M.M., Cunningham, J.P., Kaufman, M.T., Foster, J.D., Nuyujukian, P., Ryu, S.I., Shenoy, K.V. (2012). Neural population dynamics during reaching. Nature, 487(7405), 51–56.PubMedPubMedCentralGoogle Scholar
  13. Cohen, J.Y., Crowder, E.A., Heitz, R.P., Subraveti, C.R., Thompson, K.G., Woodman, G.F., Schall, J.D. (2010). Cooperation and competition among frontal eye field neurons during visual target selection. The Journal of Neuroscience, 30(9), 3227– 3238.PubMedPubMedCentralGoogle Scholar
  14. Craik, F.I., & Bialystok, E. (2006). Cognition through the lifespan: mechanisms of change. Trends in Cognitive Sciences (Regul. Ed.), 10(3), 131–138.Google Scholar
  15. Dayan, P., & Abbott, L.F. (2001). Theoretical neuroscience. Cambridge: MIT press.Google Scholar
  16. de Lafuente, V., & Romo, R. (2006). Neural correlate of subjective sensory experience gradually builds up across cortical areas. Proceedings of the National Academy of Sciences of the USA, 103(39), 14266–14271.PubMedGoogle Scholar
  17. Deco, G., Jirsa, V.K., McIntosh, A.R. (2011). Emerging concepts for the dynamical organization of resting-state activity in the brain. Nature Reviews. Neuroscience, 12(1), 43–56.PubMedGoogle Scholar
  18. Ding, L., & Gold, J.I. (2013). The basal Ganglia’s contributions to perceptual decision making. Neuron, 79 (4), 640–649.PubMedPubMedCentralGoogle Scholar
  19. Douglas, R.J., & Martin, K.A. (2007). Recurrent neuronal circuits in the neocortex. Current Biology, 17(13), 496–500.Google Scholar
  20. Dunn, F.A., & Rieke, F. (2006). The impact of photoreceptor noise on retinal gain controls. Current Opinion in Neurobiology, 16(4), 363–370.PubMedGoogle Scholar
  21. Erdos, P., & Renyi, A. (1959). On random graphs i. Publicationes Mathematicae Debrecen, 6, 290.Google Scholar
  22. Ermentrout, B. (1992). Complex dynamics in winner-take-all neural nets with slow inhibition. Neural Networks, 5(3), 415–431. ISSN 0893-6080. Scholar
  23. Faisal, A.A., Selen, L.P., Wolpert, D.M. (2008). Noise in the nervous system. Nature Reviews. Neuroscience, 9(4), 292–303.PubMedPubMedCentralGoogle Scholar
  24. Fellows, L.K. (2004). The cognitive neuroscience of human decision making: a review and conceptual framework. Behavioral and Cognitive Neuroscience Reviews, 3(3), 159–172.PubMedGoogle Scholar
  25. Fitts, P.M. (1966). Cognitive aspects of information processing. 3. Set for speed versus accuracy. Journal of Experimental Psychology, 71(6), 849–857.PubMedGoogle Scholar
  26. Fukai, T., & Tanaka, S. (1997). A simple neural network exhibiting selective activation of neuronal ensembles: from winner-take-all to winners-share-all. Neural Computation, 9(1), 77–97.PubMedGoogle Scholar
  27. Ganguli, S., Bisley, J.W., Roitman, J.D., Shadlen, M.N., Goldberg, M.E., Miller, K.D. (2008). One-dimensional dynamics of attention and decision making in lIP. Neuron, 58(1), 15–25.PubMedGoogle Scholar
  28. 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.PubMedGoogle Scholar
  29. Harvey, C.D., Coen, P., Tank, D.W. (2012). Choice-specific sequences in parietal cortex during a virtual-navigation decision task. Nature, 484(7392), 62–68.PubMedPubMedCentralGoogle Scholar
  30. He, Y., Chen, Z.J., Evans, A.C. (2007). Small-world anatomical networks in the human brain revealed by cortical thickness from MRI. Cereb Cortex, 17(10), 2407–2419.PubMedGoogle Scholar
  31. Heekeren, H.R., Marrett, S., Ungerleider, L.G. (2008). The neural systems that mediate human perceptual decision making. Nature Reviews. Neuroscience, 9(6), 467–479.PubMedGoogle Scholar
  32. Hick, W.E. (1952). On the rate of gain of information. Quarterly Journal of Experimental Psychology, 4(1), 11–26. Scholar
  33. Hodgkin, A.L., & Huxley, A.F. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. The Journal of Physiology (Lond.), 117(4), 500– 544.Google Scholar
  34. Hopfield, J.J. (1982). Neural networks and physical systems with emergent collective computational abilities. Proceedings of the National Academy of Sciences of the USA, 79(8), 2554–2558.PubMedGoogle Scholar
  35. Horn, R.A., & Johnson, C.R. (2012). Matrix Analysis. Cambridge University Press, 2nd edn.
  36. Krizhevsky, A., Sutskever, I., Hinton, G.E. (2017). Imagenet classification with deep convolutional neural networks. Commun. ACM, 60(6), 84–90. ISSN 0001-0782. Scholar
  37. Kumar, S., & Penny, W. (2014). Estimating neural response functions from fMRI. Frontiers in Neuroinformatics, 8, 48.PubMedPubMedCentralGoogle Scholar
  38. La Camera, G., Rauch, A., Thurbon, D., Luscher, H.R., Senn, W., Fusi, S. (2006). Multiple time scales of temporal response in pyramidal and fast spiking cortical neurons. Journal of Neurophysiology, 96(6), 3448–3464.PubMedGoogle Scholar
  39. London, M., Roth, A., Beeren, L., Hausser, M., Latham, P.E. (2010). Sensitivity to perturbations in vivo implies high noise and suggests rate coding in cortex. Nature, 466(7302), 123–127.PubMedPubMedCentralGoogle Scholar
  40. Luo, T., Liu, S., Li, L., Wang, Y., Zhang, S., Chen, T., Xu, Z., Temam, O., Chen, Y. (2017). Dadiannao: a neural network supercomputer. IEEE Transactions on Computers, 66(1), 73–88. ISSN 0018-9340. Scholar
  41. Maass, W. (2000). On the computational power of winner-take-all. Neural Computation, 12(11), 2519–2535.PubMedGoogle Scholar
  42. Manes, F., Sahakian, B., Clark, L., Rogers, R., Antoun, N., Aitken, M., Robbins, T. (2002). Decision-making processes following damage to the prefrontal cortex. Brain: A Journal of Neurology, 125(Pt 3), 624–639.Google Scholar
  43. Mao, Z.H., & Massaquoi, S.G. (2007). IEEE Transactions on Neural Networks, 18(1), 55–69.PubMedGoogle Scholar
  44. Markov, N.T., Ercsey-Ravasz, M., Van Essen, D.C., Knoblauch, K., Toroczkai, Z., Kennedy, H. (2013). Cortical high-density counterstream architectures. Science, 342(6158), 1238406.PubMedPubMedCentralGoogle Scholar
  45. Markram, H., Lubke, J., Frotscher, M., Roth, A., Sakmann, B. (1997). Physiology and anatomy of synaptic connections between thick tufted pyramidal neurones in the developing rat neocortex. Journal of Physiology, 500(Pt 2), 409–440. ISSN 0022-3751 (Print); 0022-3751 (Linking).PubMedGoogle Scholar
  46. Marreiros, A.C., Daunizeau, J., Kiebel, S.J., Friston, K.J. (2008). Population dynamics: variance and the sigmoid activation function. NeuroImage, 42(1), 147–157.PubMedGoogle Scholar
  47. Mason, A., & Larkman, A. (1990). Correlations between morphology and electrophysiology of pyramidal neurons in slices of rat visual cortex. II. Electrophysiology. The Journal of Neuroscience, 10(5), 1415–1428.PubMedGoogle Scholar
  48. Mason, A., Nicoll, A., Stratford, K. (1991). Synaptic transmission between individual pyramidal neurons of the rat visual cortex in vitro. Journal of Neuroscience, 11(1), 72–84.PubMedGoogle Scholar
  49. McKinstry, J.L., Fleischer, J.G., Chen, Y., Gall, W.E., Edelman, G.M. (2016). Imagery may arise from associations formed through sensory experience: a network of spiking neurons controlling a robot learns visual sequences in order to perform a mental rotation task. PLos ONE, 11(9), e0162155.PubMedPubMedCentralGoogle Scholar
  50. Melin, J. (2005). Does distribution theory contain means for extending poincaré–bendixson theory? Journal of Mathematical Analysis and Applications, 303(1), 81–89. ISSN 0022-247X. Scholar
  51. Miller, P., & Katz, D.B. (2013). Accuracy and response-time distributions for decision-making: linear perfect integrators versus nonlinear attractor-based neural circuits. Journal of Computational Neuroscience, 35(3), 261–294.PubMedPubMedCentralGoogle Scholar
  52. Munakata, Y., Herd, S.A., Chatham, C.H., Depue, B.E., Banich, M.T., O’Reilly, R.C. (2011). A unified framework for inhibitory control. Trends in Cognitive Sciences (Regul. Ed.), 15(10), 453–459.Google Scholar
  53. Patel, M., & Rangan, A. (2017). Role of the locus coeruleus in the emergence of power law wake bouts in a model of the brainstem sleep-wake system through early infancy. Journal of Theoretical Biology, 426, 82–95.PubMedGoogle Scholar
  54. Perin, R., Berger, T.K., Markram, H. (2011). A synaptic organizing principle for cortical neuronal groups. Proceedings of the National Academy of Sciences of the USA, 108(13), 5419– 5424.PubMedGoogle Scholar
  55. Platt, M.L., & Glimcher, P.W. (1999). Neural correlates of decision variables in parietal cortex. Nature, 400 (6741), 233–238.PubMedGoogle Scholar
  56. Polsky, A., Mel, B.W., Schiller, J. (2004). Computational subunits in thin dendrites of pyramidal cells. Nature Neuroscience, 7(6), 621–627.PubMedGoogle Scholar
  57. Ratcliff, R. (1978). A theory of memory retrieval. Psychological Review, 85(2), 59–108.Google Scholar
  58. Ratcliff, R., Smith, P.L., Brown, S.D., McKoon, G. (2016). Diffusion decision model current issues and history. Trends in Cognitive Sciences (Regul. Ed.), 20(4), 260–281.Google Scholar
  59. Rauch, A., La Camera, G., Luscher, H.R., Senn, W., Fusi, S. (2003). Neocortical pyramidal cells respond as integrate-and-fire neurons to in vivo-like input currents. Journal of Neurophysiology, 90(3), 1598–1612. ISSN 0022-3077 (Print); 0022-3077 (Linking). Scholar
  60. Roxin, A., Riecke, H., Solla, S.A. (2004). Self-sustained activity in a small-world network of excitable neurons. Physical Review Letters, 92, 198101. Scholar
  61. Rutishauser, U., Douglas, R.J., Slotine, J.J. (2011). Collective stability of networks of winner-take-all circuits. Neural Computation, 23(3), 735–773.PubMedGoogle Scholar
  62. Sachdev, P.S., & Malhi, G.S. (2005). Obsessive-compulsive behaviour: a disorder of decision-making. The Australian and New Zealand Journal of Psychiatry, 39(9), 757–763.PubMedGoogle Scholar
  63. Schall, J.D. (2001). Neural basis of deciding, choosing and acting. Nature Reviews. Neuroscience, 2(1), 33–42.PubMedGoogle Scholar
  64. Shadlen, M.N., & Newsome, W.T. (2001). Neural basis of a perceptual decision in the parietal cortex (area LI,P) of the rhesus monkey. Journal of Neurophysiology, 86(4), 1916–1936.PubMedGoogle Scholar
  65. Shpiro, A., Curtu, R., Rinzel, J., Rubin, N. (2007). Dynamical characteristics common to neuronal competition models. Journal of Neurophysiology, 97(1), 462–473.PubMedGoogle Scholar
  66. Sporns, O., & Honey, C.J. (2006). Small worlds inside big brains. Proceedings of the National Academy of Sciences of the USA, 103 (51), 19219–19220. ISSN 0027-8424 (Print); 0027-8424 (Linking). Scholar
  67. Taube, J.S. (2007). The head direction signal: origins and sensory-motor integration. Annual Review of Neuroscience, 30, 181–207.PubMedGoogle Scholar
  68. Thomas, N.W., & Pare, M. (2007). Temporal processing of saccade targets in parietal cortex area LI,P during visual search. Journal of Neurophysiology, 97(1), 942–947.PubMedGoogle Scholar
  69. Usher, M., & McClelland, J.L. (2001). The time course of perceptual choice: the leaky, competing accumulator model. Psychological Review, 108(3), 550–592.PubMedGoogle Scholar
  70. van den Heuvel, M.P., Stam, C.J., Boersma, M., Hulshoff Pol, H.E. (2008). Small-world and scale-free organization of voxel-based resting-state functional connectivity in the human brain. Neuroimage, 43(3), 528–539. ISSN 1095-9572 (Electronic); 1053-8119 (Linking). Scholar
  71. Wang, X.J. (2002). Probabilistic decision making by slow reverberation in cortical circuits. Neuron, 36(5), 955–968.PubMedGoogle Scholar
  72. Watts, D.J., & Strogatz, S.H. (1998). Collective dynamics of ’small-world’ networks. Nature, 393(6684), 440–442. ISSN 0028-0836 (Print); 0028-0836 (Linking). Scholar
  73. Wei, W., & Wang, X.J. (2016). Inhibitory control in the cortico-basal ganglia-thalamocortical loop complex regulation and interplay with memory and decision processes. Neuron, 92(5), 1093–1105.PubMedPubMedCentralGoogle Scholar
  74. Wilson, H.R., & Cowan, J.D. (1972). Excitatory and inhibitory interactions in localized populations of model neurons. Biophysical Journal, 12, 1–24.PubMedPubMedCentralGoogle Scholar
  75. Xie, X., Hahnloser, R., Seung, S.H. (2002). Double-ring network model of the head-direction system. Physical Review E, 66(4), 041902.Google Scholar
  76. Yamada, W., Koch, C., Adams, P. (1989). Multiple channels and calcium dynamics. In Methods in neuronal modeling: from synapses to networks (pp. 97–133). Cambridge: MIT Press.Google Scholar
  77. You, H., & Wang, D. (2017). Neuromorphic implementation of attractor dynamics in a two-variable winner-take-all circuit with nmdars: A simulation study. Frontiers in Neuroscience, 11, 40. ISSN 1662-453X. Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Swarthmore CollegeSwarthmoreUSA

Personalised recommendations