Connectivity reflects coding: a model of voltage-based STDP with homeostasis

Journal name:
Nature Neuroscience
Volume:
13,
Pages:
344–352
Year published:
DOI:
doi:10.1038/nn.2479
Received
Accepted
Published online

Abstract

Electrophysiological connectivity patterns in cortex often have a few strong connections, which are sometimes bidirectional, among a lot of weak connections. To explain these connectivity patterns, we created a model of spike timing–dependent plasticity (STDP) in which synaptic changes depend on presynaptic spike arrival and the postsynaptic membrane potential, filtered with two different time constants. Our model describes several nonlinear effects that are observed in STDP experiments, as well as the voltage dependence of plasticity. We found that, in a simulated recurrent network of spiking neurons, our plasticity rule led not only to development of localized receptive fields but also to connectivity patterns that reflect the neural code. For temporal coding procedures with spatio-temporal input correlations, strong connections were predominantly unidirectional, whereas they were bidirectional under rate-coded input with spatial correlations only. Thus, variable connectivity patterns in the brain could reflect different coding principles across brain areas; moreover, our simulations suggested that plasticity is fast.

At a glance

Figures

  1. Illustration of the model.
    Figure 1: Illustration of the model.

    Synaptic weights react to presynaptic events (top) and postsynaptic membrane potential (bottom). (a) The synaptic weight was decreased if a presynaptic spike x (green) arrived when the low-pass-filtered value ū (magenta) of the membrane potential was above θ (dashed horizontal line). (b) The synaptic weight was increased if the membrane potential u (black) was above a threshold θ+ and the low-pass-filtered value of the membrane potential ū+ (blue) was higher than a threshold θ and the presynaptic low-pass filter x macr (orange) was nonzero. (c) Step current injection made the postsynaptic neuron fire at 50 Hz in the absence of presynaptic stimulation (membrane potential u in black). No weight change was observed. Note the depolarizing spike afterpotential, consistent with experimental data. (d) Reproduced from ref. 16. (eh) Voltage-clamp experiment. A neuron received weak presynaptic stimulation of 2 Hz during 50 s while the postsynaptic voltage was clamped to values between −60 mV and 0 mV. (eg) Schematic drawing of the trace x macron (orange) of the presynaptic spike train (green) as well as the voltage (black) and the synaptic weight (blue) for hyperpolarization (e), slight depolarization (f) and large depolarization (g). (h) The weight change as a function of clamped voltage using the standard set of parameters for visual cortex data (dashed blue line, voltage paired with 25 spikes at the synapse). With a different set of parameters, the model fit the experimental data (red circles) in hippocampal slices10 (see Online Methods for details).

  2. Fitting the model to experimental data.
    Figure 2: Fitting the model to experimental data.

    (a,b) Simulated STDP experiments. (a) Spike timing–dependent learning window: synaptic weight change for different time intervals T between pre- and postsynaptic firing using 60 pre-post-pairs at 20 Hz. (b) Weight change as a function of pairing repetition frequency ρ using pairings with a time delay of +10 ms (pre-post, blue) and −10 ms (post-pre, red). Dots represent data from ref. 16 and lines represent our plasticity model. (ci) Interaction of voltage and STDP. (ce) Schematic induction protocols (green, presynaptic input; black, postsynaptic current; blue, evolution of synaptic weight). (c) Low-frequency potentiation is rescued by depolarization16. Low-frequency (0.1 Hz) pre-post spike pairs yielded LTP if a 100-ms-long depolarized current was injected around the pairing. (d) LTP failed if an additional brief hyperpolarized pulse was applied 14 ms before postsynaptic firing so that voltage is brought to rest. (e) Hyperpolarization preceding action potential prevents potentiation that normally occurred at 40 Hz16. (f) The simulated postsynaptic voltage u (black) is shown after using the protocol described in c, together with temporal averages ū (magenta) and ū+ (blue). The presynaptic spike time is indicated by the green arrow. Using the model (equation (3)) with this setting yielded potentiation. (g) Data are presented as in f but using the protocol described in d. No weight change was measured. (h) Data are presented as in f but using the protocol described in e. No weight change was measured. (i) Histogram summarizing the normalized synaptic weight of the simulation (bar) and the experimental data16 (dot, blue bar indicates variance) for 0.1-Hz pairing (control 1), 0.1-Hz pairing with the depolarization (protocol c), 0.1-Hz pairing with the depolarization and brief hyperpolarization (protocol d), 40-Hz pairing (control 2), and 40-Hz pairing with the constant hyperpolarization (protocol e); parameters are described in Table 1.

  3. Burst timing-dependent plasticity.
    Figure 3: Burst timing–dependent plasticity.

    One presynaptic spike was paired with a burst of postsynaptic spikes. This pairing was repeated 60 times at 0.1 Hz. (a) Normalized weight as a function of the number of postsynaptic spikes (1, 2, 3) at 50 Hz (dots represent data from ref. 30, crosses represent simulation). The presynaptic spike was paired +10 ms before the first postsynaptic spike (blue) or −10 ms after (red). (b) Normalized weight as a function of the frequency between the three postsynaptic action potentials (dots indicate data, lines indicate simulation, blue indicates pre-post, red indicates post-pre). (c) Normalized weight as a function of the timing between the presynaptic spike and the first postsynaptic spike of a three-spike burst at 50 Hz (dot indicates data, black lines indicate simulation). A hard upper bound was set to 250% normalized weight. The dashed line and crosses and the dotted line and stars represent simulations with alternative sets of parameters, ALTD = 21 ×10−5 mV−1, ALTP = 50 ×10—4 mV−2, τx = 143 ms, τ = 6 ms, τ+ = 5 ms and ALTD = 21 ×10—5 mV−1, ALTP = 67 ×10—4 mV−2, τx = 5 ms, τ = 8 ms, τ+ = 5 ms, respectively. Shading indicates reachable data points generated by the model with different parameters.

  4. Weight evolution in an all-to-all connected network of ten neurons.
    Figure 4: Weight evolution in an all-to-all connected network of ten neurons.

    (a) Rate code. Neurons fired at different frequencies, neuron 1 at 2 Hz, neuron 2 at 4 Hz, neuron 10 at 20 Hz. The weights (bottom) averaged over 100 s indicate that neurons with high firing rates developed strong bidirectional connections (light blue, weak connections (under 2/3 of the maximal value); yellow, strong unidirectional connections (above 2/3 of the maximal value); brown, strong bidirectional connections). The cluster is schematically represented (after). (b) Temporal code. Neurons fired successively every 20 ms (neuron 1, followed by neuron 2 20 ms later, followed by neuron 3 20 ms later, etc). Connections (bottom) were unidirectional with strong connections from presynaptic neuron with index n (vertical axis) to postsynaptic neuron with index n + 1, n + 2 and n + 3, leading to a ring-like topology. (c,d) Data are presented as in a and b, but we used a standard STDP rule12, 14, 19. Bidirectional connections are impossible.

  5. Plasticity during rate coding.
    Figure 5: Plasticity during rate coding.

    (a) A network of ten excitatory (light blue) and three inhibitory neurons (red) received feedforward inputs from 500 Poisson spike trains with a Gaussian profile of firing rates. The center of the Gaussian was shifted randomly every 100 ms (schematic network before (left) and after the plasticity experiment (right)). The temporal evolution of the weights are shown (top, small amplitudes of plasticity; bottom, normal amplitudes of plasticity; left, feedforward connections onto neuron 1; right, recurrent connections onto neuron 1). (be) Learning with small amplitudes. We used the parameters detailed in Table 1b (visual cortex) except for the amplitudes ALTP and ALTD, which were reduced by a factor 100. (b) Mean feedforward weights (left) and recurrent excitatory weights (right) averaged over 100 s. The feedforward weights (left) indicate that the neurons developed localized receptive fields (light gray). The recurrent weights (right) were classified as weak (less than 2/3 of the maximal weight, light blue), strong unidirectional (more than 2/3 of the maximal weight, yellow) or strong reciprocal (brown) connections. The diagonal is white, as self-connections do not exist in the model. (c) Data are presented as in b, but the neuron index was reordered. (d) Three snapshots of the recurrent connections taken 5 s apart indicate that recurrent connections were stable. (e) Histogram of reciprocal, unidirectional and weak connections in the recurrent network averaged over 100 s, as shown in b (fluc, fluctuations). The total number of weight fluctuations during 100 s was zero. The histogram shows an average of ten repetitions (error bars represent s.d.). (fi) Rate code during learning with normal amplitudes. We used the network described above but with a standard set of parameters (Table 1b, visual cortex). (f) Receptive fields were localized. (g) Reordering showed clusters of neurons with bidirectional coupling. These clusters were stable when averaged over 100 s. (h) Connections were able to change from one time step to the next. (i) The percentage of reciprocal connections was high, but because of fluctuations, more than 1,000 transitions between strong unidirectional to strong bidirectional or back occurred during 100 s.

  6. Temporal-coding procedure.
    Figure 6: Temporal-coding procedure.

    The same parameters were used as in Figure 5 (Table 1b, visual cortex), but input patterns were moved successively every 20 ms, corresponding to a step-wise motion of the Gaussian stimulus profile across the input neurons. (a) The schematic figure shows the network before and after the plasticity experiment. Shown are the temporal evolution of the weights (top panels: amplitude of synaptic plasticity for feedforward connections reduced by a factor of 100; left, feedforward weights onto neuron 6; right, lateral connections onto neuron 6; bottom panels: normal amplitude of plasticity; left, feedforward connections onto neuron 1; right, temporal evolution of asymmetry index of connection pattern (gray line indicates asymmetrical index for simulation; Fig. 5). Positive values indicate the weights from neurons n to n + k are stronger than those from n to n − k for 1 ≤ k ≤ 3). (b) Receptive fields are localized (left). The recurrent network developed a ring-like structure with strong unidirectional connections from neuron 8 (vertical axis) to neurons 9 and 10 (horizontal axis), etc. (small amplitudes of plasticity). (c) Data are presented as in b, but normal plasticity values were used. (d) Some of the strong unilateral connections appeared or disappeared from one time step to the next, but the ring-like network structure persisted, as the lines just above the diagonal are much more populated than the line below the diagonal. (e) Reciprocal connections are absent, but unidirectional connections fluctuated several times between weak and strong during 100 s.

  7. Receptive fields development.
    Figure 7: Receptive fields development.

    (a) A small patch of 16 × 16 pixels was chosen from the whitened natural images benchmark35. The patch was selected randomly and was presented as input to 512 neurons for 200 ms. The positive part of the image was used as the firing rate to generate Poisson spike trains of the 256 ON inputs and the negative one for the 256 OFF inputs. (b) The weights after convergence are shown for the ON inputs and the OFF inputs rearranged on a 16 × 16 image. The filter was calculated by subtracting the OFF weights from the ON weights. The filter was localized and bimodal, corresponding to an oriented receptive field. (c) Temporal evolution of the weights shown in the red dashed box in b. (d) Nine different neurons. (e) Two different neurons receiving presynaptic input with varying firing rates from 0–25 Hz (top), 0–37.5 Hz (middle) and 0–75 Hz (bottom).

References

  1. Buonomano, D.V. & Merzenich, M.M. Cortical plasticity: from synapses to maps. Annu. Rev. Neurosci. 21, 149186 (1998).
  2. Dan, Y. & Poo, M. Spike timing–dependent plasticity of neural circuits. Neuron 44, 2330 (2004).
  3. Song, S., Sjöström, P.J., Reigl, M., Nelson, S. & Chklovskii, D.B. Highly nonrandom features of synaptic connectivity in local cortical circuits. PLoS Biol. 3, e350 (2005).
  4. Lefort, S., Tomm, C., Sarria, J.C.F. & Petersen, C.C.H. The excitatory neuronal network of the C2 barrel column in mouse primary somatosensory cortex. Neuron 61, 301316 (2009).
  5. Yuste, R. & Bonhoeffer, T. Genesis of dendritic spines: insights from ultrastructural and imaging studies. Nat. Rev. Neurosci. 5, 2434 (2004).
  6. Hebb, D.O. The Organization of Behavior (Wiley, New York, 1949).
  7. Malenka, R.C. & Bear, M.F. LTP and LTD: an embarassment of riches. Neuron 44, 521 (2004).
  8. Markram, H., Lübke, J., Frotscher, M. & Sakmann, B. Regulation of synaptic efficacy by coincidence of postsynaptic APs and EPSPs. Science 275, 213215 (1997).
  9. Artola, A., Bröcher, S. & Singer, W. Different voltage-dependent thresholds for inducing long-term depression and long-term potentiation in slices of rat visual cortex. Nature 347, 6972 (1990).
  10. Ngezahayo, A., Schachner, M. & Artola, A. Synaptic activity modulates the induction of bidirectional synaptic changes in adult mouse hippocampus. J. Neurosci. 20, 24512458 (2000).
  11. Dudek, S.M. & Bear, M.F. Bidirectional long-term modification of synaptic effectiveness in the adult and immature hippocampus. J. Neurosci. 13, 29102918 (1993).
  12. Gerstner, W., Kempter, R., Van Hemmen, L. & Wagner, H. A neuronal learning rule for sub-millisecond temporal coding. Nature 383, 7681 (1996).
  13. Legenstein, R., Naeger, C. & Maass, W. What can a neuron learn with spike timing–dependent plasticity? Neural Comput. 17, 23372382 (2005).
  14. Gerstner, W. & Kistler, W.M. Spiking Neuron Models (Cambridge University Press, New York, 2002).
  15. Lisman, J. & Spruston, N. Postsynaptic depolarization requirements for LTP and LTD: a critique of spike timing-dependent plasticity. Nat. Neurosci. 8, 839841 (2005).
  16. Sjöström, P.J., Turrigiano, G.G. & Nelson, S.B. Rate, timing and cooperativity jointly determine cortical synaptic plasticity. Neuron 32, 11491164 (2001).
  17. Shouval, H.Z., Bear, M.F. & Cooper, L.N. A unified model of NMDA receptor dependent bidirectional synaptic plasticity. Proc. Natl. Acad. Sci. USA 99, 1083110836 (2002).
  18. Lisman, J.E. & Zhabotinsky, A.M. A model of synaptic memory: a CaMKII/PP1 switch that potentiates transmission by organizing an AMPA receptor anchoring assembly. Neuron 31, 191201 (2001).
  19. Song, S. & Abbott, L.F. Cortical development and remapping through spike timing–dependent plasticity. Neuron 32, 339350 (2001).
  20. Lubenov, E.V. & Siapas, A.G. Decoupling through synchrony in neuronal circuits with propagation delays. Neuron 58, 118131 (2008).
  21. Levy, N., Horn, D., Meilijson, I. & Ruppin, E. Distributed synchrony in a cell assembly of spiking neurons. Neural Netw. 14, 815824 (2001).
  22. Morrison, A., Aertsen, A. & Diesmann, M. Spike timing–dependent plasticity in balanced random networks. Neural Comput. 19, 14371467 (2007).
  23. Izhikevich, E.M. & Edelman, G.M. Large-scale model of mammalian thalamocortical systems. Proc. Natl. Acad. Sci. USA 105, 35933598 (2008).
  24. Cooper, L.N., Intrator, N., Blais, B.S. & Shouval, H.Z. Theory of Cortical Plasticity (World Scientific, Singapore, 2004).
  25. Miller, K.D. A model for the development of simple cell receptive fields and the ordered arrangement of orientation columns through activity dependent competition between ON- and OFF-center inputs. J. Neurosci. 14, 409441 (1994).
  26. Senn, W., Tsodyks, M. & Markram, H. An algorithm for modifying neurotransmitter release probability based on pre- and postsynaptic spike timing. Neural Comput. 13, 3567 (2001).
  27. Pfister, J.-P. & Gerstner, W. Triplets of spikes in a model of spike timing–dependent plasticity. J. Neurosci. 26, 96739682 (2006).
  28. O'Connor, D.H., Wittenberg, G.M. & Wang, S.S.H. Dissection of bidirectional synaptic plasticity into saturable unidirectional processes. J. Neurophysiol. 94, 15651573 (2005).
  29. Turrigiano, G.G. & Nelson, S.B. Homeostatic plasticity in the developing nervous system. Nat. Rev. Neurosci. 5, 97107 (2004).
  30. Nevian, T. & Sakmann, B. Spine Ca2+ signaling in spike timing–dependent plasticity. J. Neurosci. 26, 1100111013 (2006).
  31. Kampa, B.M., Letzkus, J.J. & Stuart, G.J. Requirement of dendritic calcium spikes for induction of spike timing–dependent synaptic plasticity. J. Physiol. (Lond.) 574, 283290 (2006).
  32. Kozloski, J. & Cecchi, G.A. Topological effects of synaptic spike timing–dependent plasticity. Preprint at <http://arxiv.org/abs/0810.0029> (2008).
  33. Jadhav, S.P., Wolfe, J. & Feldman, D.E. Sparse temporal coding of elementary tactile features during active whisker sensation. Nat. Neurosci. (2009).
  34. Blais, B.S., Intrator, N., Shouval, H. & Cooper, L. Receptive field formation in natural scene environments. Comparison of single-cell learning rules. Neural Comput. 10, 17971813 (1998).
  35. Olshausen, B.A. & Field, D.J. Emergence of simple-cell receptive field properties by learning a sparse code for natural images. Nature 381, 607609 (1996).
  36. Hyvärinen, A., Karhunen, J. & Oja, E. Independent Component Analysis (Wiley, New York, 2001).
  37. Oja, E. A simplified neuron as a principal component analyzer. J. Math. Biol. 15, 267273 (1982).
  38. Wang, H.X., Gerkin, R.C., Nauen, D.W. & Bi, G.-Q. Coactivation and timing-dependent integration of synaptic potentiation and depression. Nat. Neurosci. 8, 187193 (2005).
  39. Saudargiene, A., Porr, B. & Wörgötter, F. How the shape of pre- and postsynaptic signals can influence STDP: a biophysical model. Neural Comput. 16, 595626 (2004).
  40. Brader, J.M., Senn, W. & Fusi, S. Learning real-world stimuli in a neural network with spike-driven synaptic dynamics. Neural Comput. 19, 28812912 (2007).
  41. Clopath, C., Ziegler, L., Vasilaki, E., Büsing, L. & Gerstner, W. Tag-trigger consolidation: a model of early and late long-term potentiation and depression. PLOS Comput. Biol. 4, e1000248 (2008).
  42. Sjöström, P.J., Turrigiano, G.G. & Nelson, S.B. Neocortical LTD via coincident activation of presynaptic NMDA and cannabinoid receptors. Neuron 39, 641654 (2003).
  43. Sjöström, P.J. & Häusser, M. A cooperative switch determines the sign of synaptic plasticity in distal dendrites of neocortical pyramidal neurons. Neuron 51, 227238 (2006).
  44. Tsodyks, M.V. & Markram, H. The neural code between neocortical pyramidal neurons depends on neurotransmitter release probability. Proc. Natl. Acad. Sci. USA 94, 719723 (1997).
  45. Frey, U. & Morris, R.G.M. Synaptic tagging and long-term potentiation. Nature 385, 533536 (1997).
  46. Remy, S. & Spruston, N. Dendritic spikes induce single-burst long-term potentiation. PNAS 104, 1719217197 (2007).
  47. Hardie, J. & Spruston, N. Synaptic depolarization is more effective than back-propagating action potentials during induction of associative long-term potentiation in hippocampal pyramidal neurons. J. Neurosci. 29, 32333241 (2009).
  48. Golding, N.L., Staff, N.P. & Spruston, N. Dendritic spikes as a mechanism for cooperative long-term potentiation. Nature 418, 326331 (2002).
  49. Brette, R. & Gerstner, W. Adaptive exponential integrate-and-fire model as an effective description of neuronal activity. J. Neurophysiol. 94, 36373642 (2005).
  50. Badel, L. et al. Dynamic I-V curves are reliable predictors of naturalistic pyramidal-neuron voltage traces. J. Neurophysiol. 99, 656666 (2008).

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Author information

Affiliations

  1. Laboratory of Computational Neuroscience, Brain-Mind Institute and School of Computer and Communication Sciences, Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland.

    • Claudia Clopath,
    • Lars Büsing,
    • Eleni Vasilaki &
    • Wulfram Gerstner
  2. Present address: Institut für Grundlagen der Informationsverarbeitung, Graz University of Technology, Austria (L.B.), and Department of Computer Science, University of Sheffield, Sheffield, UK (E.V.).

    • Lars Büsing &
    • Eleni Vasilaki

Contributions

C.C. developed the model and carried out the experiments. L.B. and E.V. participated in discussions. W.G. supervised the project and wrote most of the manuscript.

Competing financial interests

The authors declare no competing financial interests.

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