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Title: Scaling Properties of Dimensionality Reduction for Neural Populations and Network Models}
Article:
Abstract:
Recent studies have applied dimensionality reduction methods to understand how the multi-dimensional structure of neural population activity gives rise to brain function. It is unclear, however, how the results obtained from dimensionality reduction generalize to recordings with larger numbers of neurons and trials or how these results relate to the underlying network structure. We address these questions by applying factor analysis to recordings in the visual cortex of non-human primates and to spiking network models that self-generate irregular activity through a balance of excitation and inhibition. We compared the scaling trends of two key outputs of dimensionality reduction—shared dimensionality and percent shared variance—with neuron and trial count. We found that the scaling properties of networks with non-clustered and clustered connectivity differed, and that the in vivo recordings were more consistent with the clustered network. Furthermore, recordings from tens of neurons were sufficient to identify the dominant modes of shared variability that generalize to larger portions of the network. These findings can help guide the interpretation of dimensionality reduction outputs in regimes of limited neuron and trial sampling and help relate these outputs to the underlying network structure.
Introduction:
Dimensionality reduction methods( for review, see[1]) have revealed compelling descriptions of neural mechanisms underlying decision-making[2, 3], motor control[4, 5], olfaction[6], working memory[7], visual attention[8], audition[9], rule learning[10], and speech[11]. These methods characterize the multi-dimensional structure of neural population activity based on how the activity of different neurons co-varies. Despite the growing use of dimensionality reduction in systems neuroscience, it is unclear whether results obtained using a limited number of neurons and trials are informative of the larger circuit from which the neurons are sampled[12]. Furthermore, since the connectivity structure of the neural population is typically unknown during in vivo recordings, it is unclear how these results relate to the underlying network structure. This paper addresses these gaps by applying dimensionality reduction to population activity from in vivo recordings and spiking network models. While our ultimate goal is to understand the population activity structure of in vivo recordings, there are several important benefits of analyzing population activity generated by spiking network models. First, because we can sample as many neurons and trials as desired from a spiking network model, we can measure how the outputs of dimensionality reduction vary over a wide range of neuron and trial counts. This allows us to assess whether the results obtained using a limited number of neurons and trials are representative of the larger network. Second, we can assess how these scaling trends are influenced by the known underlying network structure. Third, we can study how the results are influenced by which neurons are sampled in the network. This paper utilizes these three benefits of spiking network models to develop a deeper intuition for the relationship between the outputs of dimensionality reduction and the underlying neural circuit. Spiking network models that balance excitation and inhibition have been widely studied to understand the mechanisms underlying spike timing variability and correlated variability across neurons( e. g.,[13–17]). Recent studies have introduced clustering structure and found that the single-neuron and pairwise statistics of these networks better capture the slow fluctuations in firing rate observed during in vivo recordings[17–19]. In this work, we focus on two types of spiking network models: clustered networks and classic balanced( non-clustered) networks. To study single-trial population activity, we used factor analysis( FA), a linear dimensionality reduction method[2, 4, 20–22]. A key feature of FA is that it partitions spike count variability into a component that is shared among the recorded neurons and a component that is independent across neurons. From a statistical perspective, the shared component can be thought of as the co-fluctuations of the underlying firing rates of the recorded neurons, and the independent component can be thought of as the Poisson-like spiking variability of neurons[5, 23]. Whereas the independent component could be averaged away during downstream processing by pooling across neurons, the shared component can be particularly consequential for behavior[24, 25]. The partition between the shared and independent components can depend on which neurons are recorded. For example, a neuron might not co-vary with any of the recorded neurons, and instead co-vary with unrecorded neurons. By recording from more neurons, a larger proportion of a neuron’s spike count variability may be assigned to the shared component, and correspondingly less to the independent component. In this work, we leveraged this separation of variability into shared and independent components to quantify two aspects of the population activity structure:( 1) shared dimensionality, which is a measure of the complexity of the shared activity co-fluctuations, and( 2) percent shared variance, which measures the prominence of the shared component in the spiking activity. These measures generalize the ideas behind spike count correlation[26], measured between pairs of neurons, to an entire population of neurons. In addition, we used FA to identify the characteristic ways in which the neurons co-fluctuate, known as the modes of population activity. The modes of population activity have provided insight about the neural basis of working memory[27], decision making[3, 28], motor preparation[29, 30], and learning[4]. We studied the scaling trends of shared dimensionality and percent shared variance with increasing numbers of neurons and trials. To perform this analysis we used spontaneous activity recorded in the primary visual cortex( V1) of anesthetized macaque monkeys and activity generated from non-clustered and clustered spiking network models. In addition, we assessed the effects of network structure on these metrics and found substantial differences in the scaling properties of the clustered and non-clustered networks, with the clustered network showing many similarities to the in vivo recordings. Furthermore, in vivo recordings from tens of neurons were sufficient to identify dominant modes of shared variability that generalized to recordings of larger numbers of neurons. Our results demonstrate how the outputs of dimensionality reduction depend on the amount of data and the underlying network structure, and provide support for the use of dimensionality reduction with current recording technologies( i. e., tens of neurons and hundreds of trials).
Discussion:
In this study, we used V1 recordings and spiking network models to understand how the results obtained using dimensionality reduction methods generalize to recordings with larger numbers of neurons and trials, as well as how these results relate to the underlying network structure. We found that recordings of tens of neurons and hundreds of trials were sufficient to identify the dominant modes of shared variability in both in vivo recordings and a spiking network model with clustered connectivity. Comparing spiking network models, we found that scaling properties differed in non-clustered and clustered networks and that in vivo recordings were more consistent with the clustered network. These findings can help guide the interpretation of dimensionality reduction analyses in terms of limited neuron and trial sampling and underlying network structure. We focused on variability that is shared among simultaneously-recorded neurons. Shared variability has been widely studied due to its implications for the amount of information that is encoded by a population of neurons[25]. For the same population of neurons, the dimensionality computed using the raw( spike count) covariability can be substantially different from that computed using the shared covariability. To see this, consider a population of independent neurons. As the number of neurons in the analysis grows, the dimensionality based on the raw covariability would increase, whereas the dimensionality based on the shared covariability( i. e., dshared) would remain at zero because independent neurons have no shared variance. We used FA to partition the raw covariance matrix into shared and independent components and measured the dimensionality of the shared component[5, 23]. By contrast, principal components analysis( PCA), a standard dimensionality reduction method, applied to spike counts measures dimensionality of the raw covariability. Recently, Mazzucato et al. used PCA to examine the dimensionality of 3 to 9 neurons recorded simultaneously in rat gustatory cortex[19]. Despite the difference in methods used to compute dimensionality, they also found that dimensionality increases with neuron and trial count in in vivo recordings and spiking network models. Our use of FA to isolate the shared and independent components provides two important insights. First, we are able to assess the scaling trends of the dimensionality of the shared component in isolation. Relative to independent variability, shared variability is more difficult to average away within the network and is therefore more likely to influence downstream processing. Our dimensionality measurement indicates the richness of this shared aspect of the population activity. Second, we can measure the percent of the overall variance that is shared across neurons, which provides context to the dimensionality metric. For example, in the non-clustered network( Fig 6A, Top), given many trials and neurons, we identified many shared dimensions. However, these dimensions represented only a small fraction of the overall variance( Fig 6A, Bottom). By contrast, the clustered network exhibited fewer dimensions, but those dimensions represented a large fraction of the overall variance( Fig 6A). These results suggest that FA provides a more nuanced characterization of single-trial population activity than PCA. In this work, we studied spontaneous activity during in vivo recordings and in spiking network models. Our study could be extended to scaling trends in evoked activity, in which visual stimuli are presented during the V1 recordings and non-zero inputs are used in the spiking network models. Previous studies have found that shared variance tends to decrease after stimulus presentation[20, 35, 38–40] and that the scaling properties of PCA dimensionality change after stimulus presentation[19]. However, under certain conditions, the population activity patterns expressed in spontaneous activity can resemble those expressed in evoked activity[9, 41]. Studying the scaling properties of shared dimensionality and percent shared variance using evoked activity would allow us to better interpret such results in the context of limited neuron and trial sampling. We focused on trends in shared dimensionality and percent shared variance. Specific shared dimensionality and percent shared variance values obtained for the model networks likely depend on model parameters, including the number of clusters, the synaptic weights, and the probability of synaptic connection. We used the parameters described in[16], and we did not attempt to adjust parameters to make results match those found in experimental data. An interesting avenue for future work would be to understand the trade-offs among the different model parameters necessary to reproduce the absolute levels of shared dimensionality and percent shared variance measured for the in vivo recordings. While many existing models reflect various aspects of neural activity, we studied two balanced spiking network models, which can be viewed as representing the two ends of a connectivity spectrum. At one end is the classic balanced network with homogeneous connectivity which has been studied for decades( i. e., the non-clustered network)[14, 15]. At the other end is a balanced network in which each excitatory neuron belongs to a particular cluster and there is high within-cluster connectivity( i. e., the clustered network)[17–19]. Although neither of these model networks is a perfect match with cortical networks, both model networks have been shown to mimic some single-neuron and pairwise spiking statistics measured in cortical neurons[14, 15, 17, 19]. Model networks which bridge these two ends of the spectrum are currently under development. Examples include networks with spatially dependent connectivity[42] and explicit stimulus tuning structure[43]. Analysis methods similar to those used here can be applied to study the population activity structure in those networks. Comparisons between network models and in vivo recordings are usually made using aggregate single-neuron and pairwise statistics, such as mean firing rate, Fano factor, or Pearson correlation[13, 14, 17]. To move beyond single-neuron and pairwise statistics, the present work illustrates how multi-dimensional population statistics can be used to compare model networks and in vivo recordings. This approach has been adopted by several recent studies[3, 18, 19, 44] and can reveal discrepancies in the multi-dimensional activity patterns produced by model networks compared to biological networks of neurons. For example, the dominant mode of the in vivo recordings represented many neurons increasing and decreasing their activity together( Fig 4B, most elements in left-most column of the mode matrix are of the same sign). However, neither the clustered nor the non-clustered model reproduced this activity pattern in their dominant mode( Fig 8A and 8B). Such observations can guide the development of future network models. Recent developments in neural recording technology are making it feasible to record from orders of magnitude more neurons simultaneously than what is currently possible( e. g.,[45]). Thus it may soon be possible to analyze population activity for larger neuron counts from in vivo recordings. Furthermore, recent work has demonstrated the ability to access underlying network connectivity during in vivo recordings, an advance that may make it possible to determine the effects of connectivity structure on population activity[46, 47]. However, the number of trials available for studying population activity is still limited by various experimental constraints, such as an animal’s satiation or recording stability. To increase trial counts, it may be possible to combine data across multiple sessions by identifying the same neurons across multiple sessions[48–50] or by applying novel statistical methods[51–53].
We seek to understand how billions of neurons in the brain work together to give rise to everyday brain function . In most current experimental settings , we can only record from tens of neurons for a few hours at a time . A major question in systems neuroscience is whether our interpretation of how neurons interact would change if we monitor orders of magnitude more neurons and for substantially more time . In this study , we use realistic networks of model neurons , which allow us to analyze the activity from as many model neurons as we want for as long as we want . For these models , we found that we can identify the salient interactions among neurons and interpret their activity meaningfully within the range of neurons and recording time available in current experiments . Furthermore , we studied how the neural activity from the models reflects how the neurons are connected . These results help to guide the interpretation of analyses using populations of neurons in the context of the larger network to understand brain function .