Coding Capacity of Synchronous Neuronal Activity: Reliable Sparse Code by Synchrony within a Dendritic Compartment

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1 Proceedings of the th WSEAS International Conference on NEURAL NETWORKS Coding Capacity of Synchronous Neuronal Activity: Reliable Sparse Code by Synchrony within a Dendritic Compartment MÁRTON A. HAJNAL Eötvös Loránd University Department of Intelligent Systems Neural Information Processing Group Pázmány Péter sétány /C, 7 Budapest HUNGARY mahajnal@elte.hu, Abstract: Functional role of synchronous synaptic inputs to a dendritic compartment was studied. The framework of time-wise sparse coding and voltage-wise signal to noise ratios is layed out. The proportion of width of excitatory postsynaptic potential to the average period of a constantly firing input neuron define the available capacity of sparse coding in time. The reliability of transmission nonlinearly amplified if a small proportion of input neurons fire closely synchronous to each other. The asynchronous baseline, the salient synchronous peak and the theoretical maximum voltage at full synchronization determine the frame of signal to noise analysis. The available headroom between background and full sync peak and the signal to noise ratio are demonstrated to widen significantly when level of synchrony increases. Jitter reduces the signal to noise ratio and available headroom; at high synchrony level independently from input rate. Key Words: synchronization, neural code, timing, sparse code, noise, selection Introduction Synchronization is an efficient intrinsic way of inducing coherent postsynaptic responses in networks of cells; mechanism and function of oscillations are often approached separately [ ]. In this work the focus is on function. Most of the attributes of synchrony is related to the binding hypothesis [], that has gained more detailed and precise descriptions recently, such as bottom-up perceptual grouping and top-down focus of attention [9]. Parallelly processed information can be represented by different types of cell populations within a gamma cycle [2]. Even more staggering is that similar groups of cells are distinguished by their grouped phases only [7]. It is evident, that different frequency bands can operate parallelly on different representations within the same network. Subband divisions, such as high/low-γ or -β have been shown to operate on the same network parallelly. Lower level processing, or grouping-related oscillations show up at higher frequencies than attention-related oscillations [5]. The distribution of frequency bands is logarithmic. Faster rates relate to local circuitry, slower waves define interareal networks, such as theta defining the limbic system. [3]. The implications of the synchronous operating mode of small and large networks are evidently enormous. The level of study dendritic compartments as units in this paper seems unusual. In recent years, though, a new interpretation of the functional role of dendritic compartmentalization seems emerging [4]. Creation and propagation of dendritic action potentials [5] might contribute to more levels of information processing, with even plasticity present at the branch compartment level [6]. Such recent findings warrant the study of information flow in the isolated segment of a pyramidal neuron from the synapse to the proximal end of a dendritic branch compartment. It seems that some form of normalization and discretization (nonlinearity) exists already at the level of dendritic branches, potentially magnifying the computational power of at least pyramidal single neurons by orders of magnitude. The mechanisms and functions found in synaptic and whole-cell level, such as timing, impulse response and phase reset, thresholds, linear and nonlinear transfers, etc. may be present at the level between them: in dendritic branch compartments. When information propagated to more proximal dendrite branch regions, synchrony and dendritic spikes [5] enhance the propagation of timing information. Using dendritic action potentials (APs) to highlight and present salient distal information to the soma is an efficient and precise mode of operation. The activity of a postsynaptic dendrit branch, repre- ISSN: ISBN:

2 Proceedings of the th WSEAS International Conference on NEURAL NETWORKS sented by the compartment s dendritic AP, is determined by the level of synchrony vs. noise on the preceding synaptic inputs [7]. In this paper the synchron coding capacity of a theoretical dendritic branch was assessed: the scale of hundreds of inputs to a computational unit. 2 Methods Let us lay down the framework in which the coding capacity for synchronous operation is examined. Average period of firing, T =/f, is defined as the approximate reference frame for a continously firing input neuron. All postsynaptic measurements will be compared to the length of this average period. Input spike arrival time correspond to phase of excitatory postsynaptic potential (EPSP) waveform in a fixed reference period. Level of synchrony, s [, ], is defined as concentration or spread throughout the entire length of a period. So with s =, N inputs arrive spaced by an average T/N time intervals, while at s =all arrival times equal. Orifs =/3, input spikes are spread out on the first third of the period. Jitter, η, represents noise or uncertainty in timing precision of inputs. Here it is modeled as a uniform random variable between and J; J<Tbound corresponds to less than the one-period limits. Jitter is defined here in absolute terms, because the rising part of the waveforms, determinants of timing, are similarly quick with only a few ms difference between different input frequencies (see Fig. a-b). In addition, the difficulty to synchronize multiple neurons in a microcircuit or interareal network depends also an absolute timing requirements. When N inputs are spread throughout one average period of firing,..t, with s level of synchrony, the phase of the nth input is ϕ n = s +( s)n T N + η, () and η [,J] represents the random variable of temporal jitter. To simulate grouping, when only K of the total N inputs were subject to synchrony, the phase of the kth input is ϕ k = s +( s)k T + η, (2) K and the rest, for all n k with s =: T ϕ n = n + η. (3) N K To estimate the coding capacity, precise waveforms were needed to spread throughout the period. A multiple state Markov kinetics AMPA receptor ion channel model was chosen that incorporates proper rise time and the slower, two phase decay of EPSPs [8]. It is a 2-state model, expressing three open states with 2, 3 and 4 molecules of bound glutamate respectively. Desensitization is possible only for these bound forms, both from their closed and open states. The average single channel conductance of 2, 3, and 4 molecule bound open states is 5, 7, and 2pS respectively. Kinetic model parameters as fitted in [8]. To drive this channel model, a multicompartmental model of the synapse was created, with presynaptic AP, simple P/Q Ca 2+ channel, and four Ca 2+ molecule-binding fusion factor for vesicle fusion to release a pulse of glutamate into the synaptic cleft. Model was based on [9]. To study the effects of the AMPA channel current, a pyramidal spine head was modeled based on measurements of [2 26], and the voltage was recorded. The diameter of the spine head is.5μm, passive membrane resistance is 2kΩcm 2, membrane capacitance is μf/cm 2, resting potential is 7mV. To obtain EPSP waveforms not too different from voltage clamped, linear waveforms, the total AMPA synaptic conductance was a low 5pS. The model was implemented in the NEURON simulation environment. 3 Results One-period waveforms of synaptic currents (Fig. a) and steady state EPSPs (Fig. b) were obtained for different frequencies on the simulated dendritic spine endbulb. Summations of EPSPs represent dendritic integration towards the proximal end of the dendritic tree. Linear summation was considered precise, if salient synchronous waves lead to dendritic spikes, which typically do not require levels of often saturated EPSPs passively able to induce somatic action potentials [7, 5]. Fig. c illustrates summation of EPSPs at Hz with different levels of synchrony. Accompanied by jitter, even the s = total synchrony is very unlikely to reach the theoretical maximum peak voltage, to which the waveforms were normalized. As can be expected and observed in Fig. c, the more synchronous the inputs, the more lowered the baseline and pronounced both in time and voltage the peak [6]. Coding is easily interpreted imagining the following picture, measured at the AP initiation point of a dendritic compartment. The two dimensional area depicted in Fig. c, determined by length of one period and the voltage scale between the resting potential and the peak at full synchrony, is filled up by patches of ISSN: ISBN:

3 Proceedings of the th WSEAS International Conference on NEURAL NETWORKS EPSC relative amplitude 2 Hz 4 Hz Hz Hz time [ms] 2 Hz 4 Hz (a) Hz Hz time [ms].95 (b) time [ms], jitter level 2ms (c) Figure : (a) Waveforms of synaptic AMPA channel currents in the linear range (5pS conductance). (b) Waveforms of dendritic spine EPSPs. (c) Hz waveforms of N =input, spread out with different levels of synchrony (numbers on graph)..5. signal and noise. For example a highly synchronous sum of a few synaptic EPSPs is a patch of signal. A patch of noise can be an asynchronous sea of EPSPs summed to form a noisy average current (e.g. dotted s =. line in Fig. c). In this picture the width, the time scale, relates to coding capacity, the height, voltage scale, relates to reliability. A unit of information is the width of an EPSP, its relation to the period length determines the coding capacity in the framework of sparse coding [27]. The maximum available signal to noise ratio depends on the headroom, the available area above the background noise current. One set of simulations was performed with the total input synchrony measure in (). The results show some interesting features on Fig. 2. The higher the frequency, the higher the baseline and less prominent the synchronous peak. This is because the time period available to fit N input EPSPs approaches the decay time of the AMPA EPSPs as frequency increases. For the much slower NMDA receptors, with decay times at least a magnitude longer, this headroom and synchron coding capacity shrinkage for higher rates shifts even further to the lower frequencies (not shown). Regarding reliability, when a group of input neurons fire in sync with each other, their combined EPSP is much more salient than the background activity. When the synchronous firing is within a few milliseconds, the peak voltage is nonlinearly amplified due to the fast rise time of AMPA kinetics [7, 6]. Thus, effectively, a few neurons can signal a sparse code very reliably, if they fire synchronously, escaping above a much larger, noisy sea of asynchronous background. This may be the reason, why so many mechanisms are in place to control precise timing and synchrony [3]. In this study it was found that the higher the sychrony and the lower the input firing rate and the more numerous the input neurons, the more salient the combined highly synchronous, summed EPSP compared to the background (solid lines, Fig. 2). Similarly, towards the mentioned directions of parameters, the baseline decreases (dashed lines). Thus the signal to noise ratio grows, because of both the peak and the baseline movement. In addition, one of the most important results is that jitter (thin lines in Fig. 2) reduces both the signal and the available headroom. The relative contribution of absolute jitter averages out more with larger numbers of input neurons. Thus in Fig. 2c, the signal to noise and headroom- scissor closures are more visible, especially at high synchrony levels. The endpoint of the jittered peak at total synchrony is independent of frequency, and precision for signal to noise ratio is solely determined by timings. For fewer input neurons the same amount of jitter increases variability per synchron peak, thus the reliability of information propa- ISSN: ISBN:

4 Proceedings of the th WSEAS International Conference on NEURAL NETWORKS gation decreases. In order to further elaborate coding capacity, grouped simulations were performed, using K neurons in varying degree of synchrony in a background of N K neurons, Fig. 3 using (2) and (3). The observation is that the signal to noise ratio increases and headroom widens linearly with growing K/N grouping level. Lower firing rates allow for larger signals and headrooms. Jitter decreases signal peak and available headroom, again most prominently in higher synchrony and higher frequency settings. Sparse coding is defined as the average activity per total activity [27]. Sychronous sparse code in time can be interpreted as follows. Q groups, of K neurons each, represent Q numbers of synchronous signal patches along the length of a period with different phases. Define W as the effective width of larger voltage waves of one group of K synced neurons, 2-3ms for AMPA and -2ms for NMDA. T/W =/f W is the available distinguishable coding slots along the period timeline, obviously growing with decreasing frequency, since W stays constant. The sparse code in time can be represented by c = QW/T = QfW, as the number of filled slots is Q. The coding capacity depends on c combinatorically. Towards c efficiency limits the capacity [6], while at c /2 the distinguishability of two different representations [27]. Note, that besides time-wise sparsity amplified by synchrony, populational sparsity is an additional level of coding. It is conjectured from the results, that there is a common crossectional optimum of time-wise sparse coding and signal to noise constraints. 4 Discussion Traditionally the measure of uncertainty and level of dispersion in neuronal signalling is the coefficient of variation or fano factor. These measures do not take into account the electrophysiological processes of membrane voltage changes compared to the baseline. In this paper the three levels of membrane voltage has been defined to assess signal to noise ratios of synchrony information coding in periodically firing input neurons. The baseline, the salient synchronous peak and the theoretical maximum at full synchronization determine the frame of signal to noise analysis. This type of analysis is rather rare in the literature. The approach presented gives, however, insights into the nature of synchronous dendritic computation. Dendritic spikes are a compartmental level assessments of previous, distal information arriving to excitable dendritic areas. Dendritic spikes require minute precision, unless induced by continous cur- 4 Hz synchronization level, 2 input spikes (a) 4 Hz synchronization level, 5 input spikes (b) synchronization level, 5 input spikes (c) Figure 2: Signal to noise ratios and available headroom. Highest (solid) and baseline (dashed) voltage, 2ms jitter (thin), for 2, 5, and 5 input spikes. Best signals and headrooms at high synchrony and low firing rate, where, with large N, most sensitive to jitter. 4 Hz Hz Hz Hz ISSN: ISBN:

5 Proceedings of the th WSEAS International Conference on NEURAL NETWORKS 4 Hz Hz grouping percent, sync level 4 Hz Hz (a) grouping percent,.5 sync level 4 Hz Hz (b) grouping percent,.95 sync level (c) Figure 3: Signal to noise ratios and available headroom when sync-grouping a set percentage of N =5 input neurons. Highest (solid) and baseline (dashed) voltage, 2ms jitter (thin), for,.5, and.95 synchrony level. rent, which is not efficient neither in terms of information coding (only rate code is possible), nor metabolically. When synchronous EPSPs of 25-5 input neurons coalesce, the spike carries information on their number and level of synchrony combined [7]. In conclusion, if the number of synchronous inputs is significant, only those arriving in synchrony will have effect on the postsynaptic reponse. Small, individual EPSPs outside the synchronous, large, summed EPSP time-range have no effect on initiation of a dendritic spike; a form of noise filtering. The width of the time range basin is determined by input firing rates, while the width of the coding unit - the EPSP waveform - is determined by synaptic mechanisms and membrane integration time constants. It is their relative proportion that determines the coding capacity: sparse code in time. Furthermore, the dendrite compartment automatically selects out as input only those neurons that fire synchronously. It can be projected that the synchronously arriving dendritic APs to the soma have similar effect on a larger scale, the whole cell. Acknowledgements: The author was supported by the PhD program at Eötvös Loránd University. References: [] A. K. Engel, P. Fries, and W. Singer, Dynamic predictions: Oscillations and synchrony in topdown processing, Nature Reviews Neuroscience, Vol. 2, 2, pp [2] G. Buzsaki and A. Draguhn, Neuronal oscillations in cortical networks, Science, Vol. 34, 24, pp [3] P. Fries, A mechanism for cognitive dynamics: neuronal communication through neuronal coherence, Trends in Cognitive Sciences, Vol. 9, 25, pp [4] C. Borgers, S. Epstein, and N. J. Kopell, Background gamma rhythmicity and attention in cortical local circuits: A computational study, Proceedings of the National Academy of Sciences, Vol. 2, 25, pp [5] J. R. Vidal, M. Chaumon, J. K. O Regan, and C. Tallon-Baudry, Visual grouping and the focusing of attention induce gamma-band oscillations at different frequencies in human magnetoencephalogram signals, Journal of Cognitive Neuroscience, Vol. 8, 26, pp ISSN: ISBN:

6 Proceedings of the th WSEAS International Conference on NEURAL NETWORKS [6] T. J. Sejnowski and O. Paulsen, Network oscillations: Emerging computational principles, Journal of Neuroscience, Vol. 26, 26, pp [7] T. Womelsdorf, J.-M. Schoffelen, R. Oostenveld, W. Singer, R. Desimone, A. K. Engel, and P. Fries, Modulation of neuronal interactions through neuronal synchronization, Science, Vol. 36, 27, pp [8] J. Heinzle, P. Konig, and R. Salazar, Modulation of synchrony without changes in firing rates, Cognitive Neurodynamics, Vol., 27, pp [9] T. Womelsdorf and P. Fries, The role of neuronal synchronization in selective attention, Current Opinion in Neurobiology, Vol. 7, 27, pp [] K. Kitano and T. Fukai, Variability v.s. synchronicity of neuronal activity in local cortical network models with different wiring topologies, Journal of Computational Neuroscience, Vol. 23, 27, pp [] A. K. Engel and W. Singer, Temporal binding and the neural correlates of sensory awareness, Trends in Cognitive Sciences, Vol. 5, 2, pp [2] T. J. Senior, J. R. Huxter, K. Allen, J. O Neill, and J. Csicsvari, Gamma oscillatory firing reveals distinct populations of pyramidal cells in the ca region of the hippocampus, Journal of Neuroscience, Vol. 28, 28, pp [3] M. Penttonen and G. Buzsaki, Natural logarithmic relationship between brain oscillators, Thalamus & Related Systems, Vol. 2, 23, pp [4] P. Poirazi, T. Brannon, and B. W. Mel, Pyramidal neuron as two-layer neural network, Neuron, Vol. 37, 23, pp [5] N. Spruston, Pyramidal neurons: dendritic structure and synaptic integration, Nature Reviews Neuroscience, Vol. 9, 28, pp [6] A. Losonczy, J. K. Makara, and J. C. Magee, Compartmentalized dendritic plasticity and input feature storage in neurons, Nature, Vol. 452, 28, pp spikes in CA pyramidal neurons, Journal of Neuroscience, Vol. 24, No. 49, 24, pp [8] S. Raghavachari and J. E. Lisman, Properties of quantal transmission at ca synapses, Journal of Neurophysiology, Vol. 92, 24, pp [9] A. Destexhe, Z. Mainen, and T. Sejnowski, Methods in Neuronal Modeling. MIT press, Cambridge, 998, Ch. Kinetic models of synaptic transmission, pp. 25. [2] I. Segev and W. Rall, Computational study of an excitable dendritic spine, Journal of Neurophysiology, Vol. 6, 988, pp [2] M. Matsuzaki, G. C. R. Ellis-Davies, T. Nemoto, Y. Miyashita, M. Iino, and H. Kasai, Dendritic spine geometry is critical for ampa receptor expression in hippocampal ca pyramidal neurons, Nature Neuroscience, Vol. 4, 2, pp [22] D. Tsay and R. Yuste, On the electrical function of dendritic spines, Trends in Neurosciences, Vol. 27, 24, pp [23] R. Araya, K. B. Eisenthal, and R. Yuste, Dendritic spines linearize the summation of excitatory potentials, Proceedings of the National Academy of Sciences, Vol. 3, No. 49, 26, pp [24] R. Araya, J. Jiang, K. B. Eisenthal, and R. Yuste, The spine neck filters membrane potentials, Proceedings of the National Academy of Sciences, Vol. 3, 26, pp [25] R. Araya, V. Nikolenko, K. B. Eisenthal, and R. Yuste, Sodium channels amplify spine potentials, Proceedings of the National Academy of Sciences, Vol. 4, 27, pp [26] P. J. Sjostrom, E. A. Rancz, A. Roth, and M. Hausser, Dendritic excitability and synaptic plasticity, Physiol. Rev., Vol. 88, 28, pp [27] P. Foldiak, The Handbook of Brain Theory and Neural Networks, 2nd ed. MIT Press, 22, Ch. Sparse coding in the primate cortex, pp [7] S. Gasparini, M. Migliore, and J. C. Magee, On the initiation and propagation of dendritic ISSN: ISBN: