SPIKE TRAIN ANALYSIS OF SPATIAL DISCRIMINATIONS AND FUNCTIONAL CONNECTIVITY OF PAIRS OF NEURONS IN CAT STRIATE CORTEX JASON MICHAEL SAMONDS

Transcription

1 BIOMEDICAL ENGINEERING SPIKE TRAIN ANALYSIS OF SPATIAL DISCRIMINATIONS AND FUNCTIONAL CONNECTIVITY OF PAIRS OF NEURONS IN CAT STRIATE CORTEX JASON MICHAEL SAMONDS Thesis under the direction of Professor A. B. Bonds We studied changes in ensemble responses of striate cortical pairs for small (<10deg, 0.1c/deg) and large (>10deg, 0.1c/deg) differences in orientation and spatial frequency. Examination of temporal resolution and discharge history revealed advantages in discrimination from both dependent (connectivity) and independent (bursting) interspike interval properties. We found the average synergy (information greater than that summed from the individual neurons) was 50% for fine discrimination of orientation and 25% for spatial frequency and <10% for gross discrimination of both orientation and spatial frequency. Dependency (Kullback-Leibler "distance" between the actual responses and two wholly independent responses) was measured between pairs of neurons while varying orientation, spatial frequency, and contrast. In general, dependency was more selective to spatial parameters than was firing rate. Variation of dependence against spatial frequency corresponded to variation of burst rate, and was even narrower than burst rate tuning for orientation. We also found a gradual decline (adaptation) of dependency over

2 time that is faster for lower contrasts and which is likely a result of the decrease in isolated (non-burst) spikes. The results suggest that salient information is more strongly represented in bursts, but that isolated spikes also have a role in transferring this information between neurons. The dramatic influence of burst length modulation on both synaptic efficacy and dependency around the peak orientation leads to substantial cooperation that can improve discrimination in this region. Approved Date

3 SPIKE TRAIN ANALYSIS OF SPATIAL DISCRIMINATIONS AND FUNCTIONAL CONNECTIVITY OF PAIRS OF NEURONS IN CAT STRIATE CORTEX By Jason Michael Samonds Thesis Submitted to the Faculty of the Graduate School of Vanderbilt University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE in Biomedical Engineering May, 2002 Nashville, Tennessee Approved: Date:

4 ACKNOWLEDGEMENTS I express my gratitude to Professor A. B. Bonds for his guidance and encouragement throughout this project, as well as his support during my time at Vanderbilt University. I would also like to extend my gratitude to Professor Don Johnson for his assistance in explaining the finer details of type analysis, and to Professor Jonathan Victor for sharing his knowledge and experience with spike train analysis. I am very grateful to Professor Ross Snider for working with me in order to use his spike sorting and cross-correlation software to contribute to my results. And lastly, I would like to thank Dr. John Allison and Heather Brown for their help in collecting the data. Although this project would not be possible without their assistance, all the ideas, type analysis software, writing, and conclusions are my own work. I would also like to thank the Graduate School and the National Institute of Health (Grant RO1 EY03778) for providing financial support during my time in graduate school. And it goes without saying that I am always grateful for the support from friends and family that has always been there throughout my educational pursuits. ii

5 TABLE OF CONTENTS ACKNOWLEDGEMENTS...ii LIST OF FIGURES... iv Chapter I. NEURAL POPULATION ANALYSIS REVIEW... 1 Page Introduction... 1 Theoretical Background... 2 Single-unit Research... 6 Multi-unit Research Functional Imaging Correlation and Connectivity Point Process and Cross-correlation Partialization Gravitational Clustering Information Theory: Dependency and Complexity Causality Nonlinear Methods Neural Code Theory Average Spike Rate Code Temporal Code Bursting Latency Spatiotemporal Patterns Oscillations Chaos Theory Information Theory The Future II. COOPERATION BETWEEN AREA 17 NEURON PAIRS THAT ENHANCES FINE DISCRIMINATION OF ORIENTATION Introduction Methods Preparation Stimuli Data Acquisition and Spike Classification Type Analysis iii

6 Results Latency Temporal Resolution Discharge History Synergy, Independence, and Redundancy Confidence in Distance Estimations Functional Connectivity Discussion Latency Independent ISI Characteristics Bursts and Connectivity Functional Connectivity and Synergy Orientation Discrimination III. FUTURE EXPLORATIONS Introduction Cortical Function Theory Multidimensional Data Cortical Clustering Spatiotemporal Connectivity REFERENCES iv

7 LIST OF FIGURES Figure Page 1. An example of firing rate tuning and fine and gross discriminations Enhanced discrimination with latency differences Distance rates versus temporal resolution Determination of Markov order of analysis Discharge history contribution to orientation discrimination Discharge history contribution to spatial frequency discrimination Ensemble distance versus individual neuron distances Distances and synergy versus random sample size Another example of sample size functions Temporal dynamics of dependnecy between neurons Difference between dependency and firing rate during contrast modulation Temporal dynamics of dependency adaptation Dependency tuning for orientation and spatial frequency modulation v

8 CHAPTER I NEURAL POPULATION ANALYSIS REVIEW Introduction One of the most elusive questions in biological sciences has been how the brain is able to encode and decode the multi-dimensional features of sensory signals and to formulate perceptions or perform actions on the order of hundreds of milliseconds. The majority of neurophysiological studies have relied on counting the number of spikes recorded from a single neuron and demonstrating how this spike count varies according to sensory input features. Although these data represent the average response of only one neuron to specific stimulus features, neurophysiologists have at the same time acknowledged that the brain could only perform their complex operations with populations of neurons. The brain cannot be thought of as simply as a passive screen receiving a projected image of the outside world. The brain is able to separate figure from background, perform invariant recognition, and make accurate and generalized predictions. Understanding the representation of sensory information both on local and global levels is equally important, and neither of these tasks presents a clear approach to finding a solution. It cannot even be posed as a simple encoding and decoding problem because there is not always a clear input and output. In this introduction, we describe how the theories of brain function have evolved over the last two centuries and how anatomical and physiological studies of individual neurons and small groups of neurons have contributed to these theories. Then we 1

9 describe the methods that have been developed to analyze interactions between nerve cells and when and how they are applicable to providing insight to cortical functions. One of the difficulties in understanding brain function is deciding where to start the analysis. We describe some of the theories of what aspects of the individual neuron's signal carry the sensory information. Since we are looking at how information is transmitted and manipulated in the brain, information theory has become an important analytical tool. Lastly, we discuss where the theories of neural population codes might be heading and how some of the conflicting arguments might be resolved. Theoretical background In a recent review, Doetsch (2000) points out that the idea that sensory information was encoded in patterns of populations of neurons was proposed as early as 1802 by Young and elaborated by von Helmholtz in 1860 for explaining color vision. Sherrington (1941) wrote on the importance of understanding the cooperation of groups of nerve cells beyond their individual properties even with the understanding that the brain does utilize localization for many of its functions. In 1949, Hebb suggested that groups of cells form regional circuits and are activated by the appropriate spatiotemporal firing pattern and then produce some appropriate spatiotemporal output pattern. Hebb s theory was proposed to explain many of the phenomena observed in psychophysical studies. The main idea is that neurons that fire together will wire together, which is the foundation behind learning in the brain. Learning itself is a slow and tedious process to create the wiring that later on leads to the fast and generalized perceptions. 2

10 In 1972, Barlow proposed a contradictory theory where the individual cells represent their information independently. The theory was based on the data that was available at the time from single neuron (single-unit) recordings. The idea is that the individual nerve cells are very specialized feature detectors that are activated with the appropriate stimulus in the appropriate location. The theory is known as the cardinal cell theory because the information converges as it is passed on to more specialized cells higher up in the hierarchy of the brain s perceptual regions. However, the convergence is not so drastic as to end up activating a single grandmother cell (the notion that every perception has its own cell i.e., your grandmother activates a particular nerve cell). Von der Malsburg (1981) introduced a correlation theory based on some of the Hebbian principles, along with theories on pattern recognition and neural networks. The purpose of this theory was to address the deficiencies of the previous brain function theories and propose solutions to these problems. In general, von der Malsburg s correlation theory is based on the synaptic strength modulations on short- and long-term time scales (also a theory behind short- and long-term memories). The long-term modulations are based on anatomical and physiological modifications of synaptic connections, while the short-term modulations might be induced within the temporal structure of cellular signals. The key is that synaptic strengths are dynamic and lead to competition and the creation of subnets within a larger network. Uncorrelated subnets can coexist without interference and it is correlation (or synaptic coupling) that ties information together and determines the activity patterns. Rather than requiring hardwired specialized detectors, von der Malsburg s theory predicts that only simple feature 3

11 detectors would be required and that more complex features are extracted through the activation of synaptic subnets. Another theory derived from the combination of theoretical neural networks and neurophysiological data is the synfire chain model of the cortex (Abeles 1991). The synfire chain model is a network of converging and diverging connections where synchronization is fundamental to the processing and transmission of neural information. An individual neuron basically acts as a coincidence detector (Abeles, 1982) that passes on spikes to the postsynaptic neuron that are synchronized with sub-millisecond precision. One of the motivating aspects behind the theory is that the neuron is typically much more sensitive to synchronous inputs over integrating (asynchronous) inputs. An argument against this theory is the unreliability of synaptic transmission (Shadlen and Newsome, 1994), but theoretical models have demonstrated the possibility of submillisecond precision with a synfire chain model incorporating on the order of 100 neurons (Deismann et al., 1999). Shadlen and Newsome (1994) believe the unreliable synapse can only be used in a model that incorporates integration. Each neuron receives thousands of inputs that create a balance between excitation and inhibition (chaos) to reach threshold in the postsynaptic neuron faster than with the resting membrane potential, but still avoid saturation. The signals of individual neurons are very noisy and highly redundant. They believe the sensory information is represented by the form of an average firing rate pooled across populations of neurons. There is an asymptote reached for signalto-noise with populations of this size as averaging cannot remove correlated noise. The irregularity of the interspike interval (ISI) would seem to be an argument for its role in 4

12 carrying neural information, but Shadlen and Newsome (1998) believe that this irregularity is a result of inhibition in balancing the chaos. They believe that redundancy is not a problem with the massive number of neurons in the cortex and reasonably accurate firing rates can be transmitted with integration times as fast as 10 ms. Hopfield (1995) views the problem of understanding brain function by considering the problem of pattern recognition and the capabilities of neurophysiology. Although it might not be as efficient from an information-theoretic point of view, a code based on timing rather than rate makes more sense from both the pattern recognition and biological point of view. A network based on timing allows for scale-invariant recognition. Delays can easily be caused by synaptic, axonal, or cellular mechanisms and decoding is provided by a coincident-detection scheme. The physiological and psychophysical studies of the last 20 years have continued to demonstrate the nonlinearities of the visual system (Wilson and Wilkinson, 1997). The nonlinear pooling mechanisms and interactions that are found in the neural processing suggest a model similar to winner-take-all (WTA) networks and make it clear that the visual system cannot be broken down into independent spatial channels. These mechanisms have been demonstrated for texture perception, stereopsis, motion perception, and form perception. Wilson and Wilkinson (1997) point out that one of the reasons nonlinear mechanisms would make sense for visual processing is that no matter how many linear calculations are performed, they can always be reduced down to one single linear calculation. Lastly, an alternative to neural network-based or spatiotemporal-based coding models is a population coding theory where individual neurons can be thought of as 5

13 vectors (Pouget et al., 2000). Because individual neurons are tuned to a feature, the magnitude of the responses of a population of neurons can be added in vector space based on their tuning properties. The theory has been derived from populations in the middle temporal visual area (area MT) and motor cortical studies to determine direction. The vector approach allows for nonlinear mappings and is most efficient when applying Bayesian classification principles. However, recent evidence against the population vector hypothesis has been shown in the neural activity of the motor cortex in predicting hand movement (Scott et al. 2001). Single-unit research The brain averages responses across populations whereas in the laboratory it is averaged over time. This practice is based on the assumption that the average firing rate is the primary component in the representation of sensory information and that the single unit is sufficiently representative of the population. Still, regardless of the theory of brain function, single-unit recordings can be used to reveal certain properties of the network circuitry with careful selection of the stimulus, iontophoretic application of neurotransmitters, and anatomical studies of the cell types and synaptic connections at the recording sites. Hubel and Weisel (1962) were the first to study the functional architecture of the visual cortex through an extensive analysis of the individual neurons. The anatomy of the visual cortex reveals 6 distinct layers and diversity and organization of cells, but Hubel and Weisel demonstrated organization and cell types by measuring the responses of single units. They discovered visual cortical cells responded to bars of light at a 6

14 preferred orientation, sometimes at a preferred direction, and across a continuum of monocular to binocular stimulation. Their research demonstrated an organization of orientation columns and ocular dominance hypercolumns across the cortical layers, along with a retinotopic mapping across the surface of the cortex. The analysis of the receptive field properties of the cells revealed two primary classifications of cells (simple and complex). They proposed a feed-forward linear model of the visual cortex based on the properties of lateral geniculate nucleus (LGN) and cortical cells. The model proposes a hierarchy from LGN to simple to complex cells that explains the origin of orientation tuning, along with the other receptive field properties they discovered. This same experimental approach was used more recently to examine functional organization of additional properties of cortical cells (DeAngelis et al., 1999). DeAngelis et al.'s results showed that although almost all properties (spatial frequency, orientation, temporal frequency, and latency) were organized in clusters and columns, there was diversity in the organization in the spatiotemporal receptive fields. The difference in spatial phase of the receptive fields of nearby cells prevents overlap and redundancy among the clusters and columns. Their results provide an explanation for the relative lack of redundancy found in nearby cortical neurons when the tuning characteristics would suggest otherwise. Sillito (1972) studied the role inhibition had in receptive field properties by iontophoretically applying bicuculline to suppress the inhibitory neurotransmitter gamma-aminobutyric acid (GABA). By comparing tuning functions of individual cells with and without inhibition, Sillito was able to demonstrate that inhibitory mechanisms play a role in simple and complex cell orientation tuning. Without inhibition, the simple 7

15 cells had much broader tuning, lost the linear on and off receptive field distinctions, and had a loss or reduction in directionality specificity. Bicuculline resulted in less dramatic broadening or no change in orientation tuning and a less significant effect in directionality specificity for complex cells. The results provide evidence that the visual cortex cannot be thought of as a simple linear feed-forward model (Hubel and Weisel, 1962). They do not rule out the role feed-forward mechanisms might have in receptive field properties such as orientation tuning, but simply demonstrate that inhibitory mechanisms also play a role and a more complicated model of cortical organization is required. Toyoma et al. (1974) examined the organization of the visual cortex by using a stimulating electrode along with a recording electrode. This procedure allowed them to examine axonal projections and synaptic connections within and across cortical layers. After determining the organization of the projections and identifying excitatory and inhibitory connections, they were able to come up with a rough model of the circuitry of the cortex. Even their simple model once again demonstrated the influence of inhibition and the complexity of the cortical network with the inclusion of inhibitory interneurons. Creutzfield et al. (1974a,b) examined the vertical organization of the visual cortex with intracellular recordings and analysis of the peristimulus time histogram (PSTH). They found that inhibition usually followed an excitatory response and that orientation tuning did not appear to be a simple result of precise spatial arrangement from afferent neurons. They also did not find a lot of shared input or any excitatory connections within orientation columns suggesting there is not a lot of convergence. The inhibition they found was not also very localized, but was almost always apparent in a diffuse form. 8

16 Their results suggest that many of the large number of synaptic connections within the cortex are inhibitory. Another method used to derive network properties from the response of a single unit is to use sub-threshold stimulation. Sub-threshold stimulation is when a stimulus does not evoke a response when presented alone, but causes a change in the response to another stimuli that does evoke a response. Because the sub-threshold stimulation is below threshold it does not result in a response on its own, but it does still induce a postsynaptic potential, which can lead to changes in the network interactions when stimuli are shown that do result in a response. One example of this protocol is the cross-orientation stimulus (Morrone et al., 1982; Bonds, 1989) used to study the role of inhibition in orientation tuning. The stimulus consists of two rapidly interleaved sine wave gratings with one grating at the optimal orientation and the other one varied to reduce the response. The results of these two studies suggested that the inhibition was a result of pools of cells and not a property of the recording cell. The results also suggest that inhibition is intracortical (from other simple or complex cells and not from LGN cells). Another example of a sub-threshold stimulus is stimulation outside of the classic receptive field. The term classic receptive field is used because the receptive field was traditional referred to as the region in the visual field which when stimulated produced an excitatory response. Sillito and Jones (1996) used both discrete stimuli and an annulus outside of the classic receptive field while stimulating the classic receptive field and find facilitation many times with cross-oriented stimulation in the periphery, suggesting a possible "discontinuity detector" and at least demonstrating further complexities of 9

17 neurons when not considered in the context of the network. Vinje and Gallant (2000) have also used stimulation outside of the class receptive field to verify their natural stimulus results that suggest the cortex employs a sparsely distributed representation. Single-unit responses can also be analyzed across time to provide some clues into the population dynamics. Volgushev et al. (1995) studied the postsysnaptic potential (PSP) responses and found that excitatory orientation tuning becomes tighter over a ms period. Their results suggested that a ms delayed (likely feedback) inhibition played a role in the narrower tuning. Ringach et al. (1997) found similar results using a reverse correlation method. However, they found that the delayed suppression had broader tuning than the excitation and that the overall sharpened tuning occurred within 6-10 ms. Rolls et al. (1997a) analyzed a population of 14 neurons individually recorded in the inferior temporal cortex (IT) for responses to 20 visual stimuli. Because the recordings were not simultaneous, the analysis ignores any temporal dependencies between neurons. The cells they recorded are involved in face recognition and they used information-theoretic approaches to determine the redundancy or independence of the neurons. Even though they could not document any interactions between cells, they still found the neurons to be relatively independent and that the representation of faces was distributed in IT. Multi-unit research As early as 1981, a population of 19 neurons was recorded simultaneously in the monkey visual cortex using a 30-electrode microelectrode array (Kruger and Bach, 10

18 1981). In the last two decades, there have been advances in the areas of microelectrode arrays and tetrode arrays, but it is still difficult to obtain high resolution simultaneous recordings of greater than 100 neurons (Nadasdy, 2000). With the improvements and availability of this technology more research has moved into the area of population analysis and has stimulated many recent reviews (Pouget et al., 2000; Milton and Mackey, 2000; Doetsch, 2000; Nadasdy, 2000). Studies are beginning to show the ability to understand the neural code from simultaneous population recordings (greater than pairs) in the aplysia abdominal ganglion (Wu et al., 1994), the rat motor cortex (Laubach et al., 2000), the primate motor cortex (Maynard et al., 1999; Wessberg et al., 2000), the moth olfactory lobe (Christensen et al., 2000), the somatosensory cortex (Doetsch, 2000; Nicolelis et al., 1997), the rat hippocampus (Nadasdy, 2000), the retina, (Warland et al., 1997), the auditory cortex (Eggermont, 1998), the LGN (Mehta et al., 2000), and the visual cortex (Gray et al., 1995; Nordhausen et al., 1996; Reich et al., 2001). Rolls et al. (1997) point out that even a sparsely distributed representation would have drastic advantages in efficiency of encoding. If encoding were done on a single neuron level, the number of representations would be equal to the number of neurons. If the encoding were fully distributed, the number of possible representations would be equal to 2 raised to the number of neurons (2 #neurons ). Results do appear to support the idea that responses are in some way distributed across cortical regions. Even without the advances of multi-neuronal (multi-unit) recording technology, studies have been done on small populations of neurons (usually on pairs of neurons) to reveal properties of the cortex as a network. This is possible with the use of a single electrode and spike sorting algorithms (Abeles and Goldstein, 1977; Snider and Bonds, 11

19 1998) or two electrodes recording simultaneously. The recordings of small populations within a small region are then analyzed for correlation and functional connectivity. Cross-correlation analysis has been used in the visual cortex to study the connections and organization within and across cortical layers (Toyoma et al., 1981a,b; Michalski et al., 1983; Alonso and Martinez, 1998). Toyoma et al. (1981a,b) used the neurotransmitter glutamate to enhance their responses and found that half the pairs of cells they recorded shared common input and only 10% of the pairs showed any direct excitatory or inhibitory interaction. They found common excitatory input connections into layer III to V (likely from LGN), intracortical direct excitatory connections from layer III-IV to layer II-III, and intracortical inhibitory direct connections from the deeper part of layer IV up to the middle layers. Inhibition was found to be between simple cells or from simple cells to complex cells, and only excitation was found between complex cells. They also did not find many direct connections across orientation columns. Michalski et al. (1983) found similar results with rare connections across columns and found twice as many direct excitatory over direct inhibitory connections within columns. Alonso and Martinez (1998) were able to find more direct excitatory connections between layer IV simple cells and layer II/III complex cells, but also reported a continuum of shared input to direct connections from layer IV to layer II/III demonstrating that the LGN does not only project into layer IV and providing further evidence against the feed-forward hierarchical model. Cross-correlation has also been used to verify long-range connections (>1mm) in the cortex (Ts o et al., 1986). Ts o et al. found excitatory interactions across several millimeters using two electrodes. The correlation was most apparent when the two cells 12

20 had similar orientation preferences and facilitation was found when the cells had similar eye preferences. Gray et al. (1989) have also examined long-range interactions between cells and discovered that cells that had oscillatory responses (40-60 Hz) that were precisely synchronized. The synchronization was strongest when stimuli had similar orientations and in the same direction, and even stronger with a single object to stimulate both cells. Singer and Gray (1995) have proposed that the oscillations are a mechanism for long-range synchronization and that it might have a role in either synchronizing cell assemblies or binding features of an object (because it is strongest with coherent and connected stimuli). Information-theoretic analysis of small populations of neurons has also provided evidence on the redundancy, independence, or cooperation between neurons. The results have been used to provide support for or against brain function theories. Warland et al. (1997) analyzed populations of retinal ganglion cells and found the information to be redundant unless cell types differed and even then the maximum advantage of information as a population was reached at 4 cells. Nirenburg et al. (2001) also studied retinal ganglion cells in pairs and found that ignoring the correlation between the cells still provided over 90% of the possible information suggesting that cells for the most part act independently. Dan et al. (1998) studied pairs of cells in the LGN and found that the precise synchronizations provided on average an additional 20% more information. Gawne et al. (1996a) showed that on average 20% of the information in nearby visual cortical cells was redundant, and Reich et al. (2001c) found the information to be independent in the visual cortex unless the responses were summed (where useful information may be discarded). 13

21 Functional imaging Alternatives to electrophysiological recordings, such as functional magnetic resonance imaging (fmri), positron emission tomography (PET), and optical imaging, can also be used to reveal population activity, although it is not able to reveal anything about the underlying code. Functional imaging is able to provide localization of brain activity by measuring changes in the hemodynamic response, but is unable to provide accurate temporal information and information about the individual cellular responses. As fmri and functional PET studies continue to grow and a better understanding of how the signal relates to cellular activity is achieved, the results can continue to aid in the understanding of neural processing (Raichle, 1998). Very recently fmri responses have been compared directly to neural spiking responses (Logothetis et al., 2001), where the results showed that the hemodynamic response may underestimate neural output because of the lower signal to noise ratio found in fmri. However, it may also overestimate activity because it was found that responses are linked to incoming input and local responses and not the output activity (i.e., high synaptic activity does not necessarily result in high output activity). One of the most appealing aspects of functional imaging is that it provides a non-invasive measurement of neural activity that can be used to compare human responses with animal neurophysiological data. Optical imaging has provided better spatial resolution than fmri or functional PET, but does have limitations because it records only surface hemodynamics. It has been successful in displaying the functional architecture of the upper layers of the visual cortex by showing the orientation columns clearly and discovering their pinwheel 14

22 organization (Grinvald, 1992). Optical imaging has been drastically improved over the last decade with the temporal precision of recording to go along with the spatial precision by using voltage-sensitive dyes (Fitzpatrick, 2000). Correlation and Connectivity Point process and cross-correlation In (1967a), Perkel et al. introduced the study of neuronal spike trains in terms of stochastic point processes. When looking at the information-bearing aspect of neuronal spike trains, the importance is in the times at which discharges occur and not in the precise voltage measurements or the variations in the action potential waveforms. A stochastic point process consists of a series of point events that are considered instantaneous and indistinguishable. By analyzing spike trains as stochastic point processes, it allows the investigator to implement many computational techniques that will allow them to extract information about the function and the mechanisms of the nervous system. Careful study of the temporal relationships in an observed cell can reveal how the cell produces spikes and how a presynaptic input is transformed into a postsynaptic output. More importantly, looking at multiple spike trains simultaneously recorded provides the information necessary to understand the interconnections and functional interactions between cells. The statistical analysis of pairs of neuronal spike trains was the genesis of the study of brain function in terms of groups of neurons. Extending the approach of expressing neuronal spike trains as stochastic point processes, Perkel et al. (1967b) introduced a method of statistical analysis for two 15

23 simultaneously recorded spike trains. Measuring the backward and forward recurrence times of spikes from one neuron relative to each spike in the other neuron creates a crosscorrelation function. The cross-interval histogram takes spikes from one train and is a histogram of the times to the nearest spikes in the other train. The cross-correlation histogram is a histogram of all the spikes in one train to each spike in the other train. The cross-interval histogram is used to corroborate independence indicated by the crosscorrelation histogram or to explore suspected short-latency interactions. The cross-correlation histogram is used to detect possible dependencies between a pair of neurons. This dependence can result from either (or both) the functional interaction between the two neurons or from a common input. The interaction can be a result of direct synaptic connection or mediated through interneurons. One difficulty discovered with the cross-correlation histogram is the ability to determine independence when looking at pacemaker cells. The cyclic action of the cells can lead to false designation of dependence between cells when in fact they are independent. There can also be false attributions of independence when the dependence is too weak to be noticed above noise levels. Another problem of applying cross-correlation analysis to neurophysiological experiments is that cross-correlation histograms will detect changes in firing rate as dependencies. This can be difficult to avoid because of response changes that occur naturally during experiments. Moderate degrees of nonstationarity, however, will not mask out effects when there is neuronal interaction. Computer simulations were used to produce cross-correlation histograms to be used as templates or rules for classifying experimental data (Perkel et al., 1967b). The simulations showed that there are difficulties in discriminating common input 16

24 dependencies and indirect connection dependencies. Several different arrangements of functional interaction can lead to the same cross-correlation. It is important to remember when using cross-correlation analysis that the results provide insight as to possible connections and interactions and do not represent any information on the actual anatomy or physiology. The cross-correlation histogram can be used to distinguish neuronal pairs between three different functional relationships: (1) no interaction, (2) interaction (either direct or through interneurons), and (3) stimulus-modulated interaction (the interaction is modified by the stimulus). The functional relationships are determined from the cross-correlation histogram and the prediction of the cross-correlation. A prediction of the crosscorrelation can be determined with the mean firing rates of both cells under stimulated and unstimulated conditions, the cross correlation function with the stimulus off, and the PSTHs for both cells. The predicted cross-correlation is used to determine whether the interaction between neurons is stimulus-modulated. The rules for determining functional relationship are: If the cross-correlation histogram is flat, there is no interaction. If the cross-correlation histogram does agree with the predicted crosscorrelation, the interaction is not stimulus-modulated. If the cross-correlation histogram does not agree with the predicted crosscorrelation, the interaction is modulated by the stimulus. Gerstein and Perkel added a new dimension to cross-correlation analysis in 1972 by introducing the joint PST scatter diagram. The joint PST scatter diagram is essentially another method of displaying the correlation between spike trains, but provides greater 17

25 understanding into the interactions between the neurons. The scatter diagram is created by plotting the spike train of one neuron versus the spike train of the other neuron. A point is plotted wherever an occurrence of a spike from one neuron crosses the occurrence of a spike from the other neuron. As cross-correlation analysis has been incorporated into neurophysiology studies, there have been observations made to better describe the properties of the techniques and changes made to improve the techniques. One property of cross-correlation analysis that was discovered was an asymmetry in the sensitivity of cross-correlation analysis for excitatory versus inhibitory interactions. Aertsen and Gerstein (1985) discovered through an evaluation of neural connectivity and the associated cross-correlation analysis that it is much more difficult for inhibitory effects to appear in the cross-correlation histogram than excitatory effects. Unless the inhibitory effects are significant, they will go unnoticed during the cross-correlation analysis. Because of this asymmetry, there may be a false indication of more excitatory connections than inhibitory connections occurring in different studies using cross-correlation analysis. Enhancements were made by Palm et al. (1988) and Aertsen et al. (1989) to the cross-correlation analysis techniques to provide a quantitative approach to classifying neuronal interactions. Formulae for probability distributions of measures were created so that data could be compared under significance levels. This allows inferences to be evaluated using a significance test referred to as surprise. A quantification procedure was created for the study of stimulus-locked, time-dependent correlation of firing between two neurons so that direct and indirect stimulus effects could be described quantitatively. The changes make it easier to determine effective connectivity. 18

26 Quantitative measures can be used to separate characteristics of diagonal features to determine whether interaction results from direct interaction or shared input. The additions to the cross-correlation analysis still only determine effective connectivity, which is not necessarily the actual anatomical description of the connections. It should be thought of as an equivalent neuronal circuit that can represent any number of actual physiological circuits that would result in the same output. Partialization Because connections between any 2 neurons in the visual cortex are usually weak, it is usually difficult to detect the coactivation of groups of neurons because of the complicated circuitry in between the 2 neurons. An alternative to the shift predictor method (Perkel et al., 1967) is the method of partialization. Partialization is used in conjunction with cross-correlation. It separates out the independent and common input contributions between two neurons in the Fourier domain to make an estimate of the functional connectivity. The method is more effective as the population of assemblies grows and has been successfully used to analyze changes in assembly strength with respect to changes in anesthesia (van der Togt et al., 1998). Because the effectiveness of partialization depends on larger populations of assemblies, the method is not advantageous over shift predictor methods when analyzing pairs of neurons. Gravitational clustering Advances in techniques allowing larger populations of neurons to be simultaneously recorded have led to a new approach in the analysis of these populations. 19

27 In 1985, Gerstein and his colleagues describe a method that analyzes groups of neurons as a whole rather than in pairs. Simultaneous recordings of 10 or more neurons would be very difficult using cross-correlation analysis because the relationships have to be examined pairwise. Their new approach maps the activity of neurons into motions of particles in a multidimensional Euclidean space. Each neuron is thought of as a point particle in space and each spike results in an increment in a charge for that particle. The particles are then essentially plotted in space and by observing their movements, the interrelationships between the neurons can be determined. The neurons that are interconnected tend to move towards each other so after a simulation, groups of neurons that are connected or receive the same input will cluster together. Relationships of the neurons can also be seen by plotting the distance between pairs of neurons versus time. The stronger the connection between neurons, the faster the distance will approach zero. If the neurons are independent from each other, the distance will remain constant. There are many variations of this technique that can factor into the effectiveness of this approach to describing neuronal group characteristics. Modifying the definition of the charges and force rules are necessary in order to observe inhibitory relationships. The approach does appear to display successfully the characteristics of a simulated group of 10 neurons. Only 50 spikes from each neuron were necessary to produce the clusters and display the relationships within the network. Compared to the cross-correlation techniques that require hundreds to thousands of spikes to demonstrate similar results, this approach appears to be much more sensitive (Gerstein et al., 1985; Gerstein and Perkel, 1985; Strangman, 1997). The method has been successfully applied in several neurophysiological studies (Lindsey et al., 1992a,b; Lindsey et al., 1994; Maldonado and 20

28 Gerstein, 1996; Lindsey et al., 1997; Morris et al., 2001) and has recently been improved to detect weak synchrony among neural populations at various spike intervals (Baker and Gerstein, 2000). Because the results of gravitational clustering are not as clear as cross correlation, the method is typically only used for studying larger populations of neurons. Information theory: dependency and complexity Another method recently developed under information-theoretic principles can be used to compare the probability distributions of neurons and the temporal dynamics of their dependence (Johnson et al., 2001). These analysis techniques can be carried out on larger populations of neurons or on a pair-by-pair basis to determine the neuronal dependency of a network and how the dependency changes across time and across stimulus modulations. The method forms probability mass functions (types) for spatiotemporal patterns and then calculates the probability functions if the neurons were assumed to be independent (forced-independent type). An accumulated distance between the two responses is then calculated across time. If the neurons are independent, their probabilities of firing at given time in a spatiotemporal pattern should be equal to the product of their individual probabilities of firing. Any variance from this equality means there is some inhibitory (less than the forced-independent) or excitatory (greater than the forced-independent) dependency between the individual neurons. Tononi et al. (1994) also developed an information-theoretic method to measure connectivity. Their method measures the deviance from independence from entropy and mutual information calculations. The complexity is then defined as the relative deviance of a local region with respect to the deviation from the average deviance of the overall 21

29 network. The complexity of a network is lowest when the units are fully integrated or when the units are fully independent, and the complexity is highest in between the two extremes (smaller strongly connected groups sparsely connected). The method can be used in functional imaging studies where the voxel or pixel represents the individual unit and the results can characterize complexity changes when the strength of activity does not vary, which is the case in pathologies such as schizophrenia (Tononi et al., 1997). The method can be applied to any multi-dimensional data set making it ideal for neurophysiological studies as well as neuroimaging studies. Beyond identifying the strength of complexity, the method has been further expanded to characterize the complexity (Sporns et al., 2000). In other words, the functional clusters can be identified so that connectivity patterns can be identified and related back to behavioral changes. Causality A problem with correlation and coherence measurements is that many times they do not resolve the directionality of information flow, which becomes very relevant in the brain with both feed-forward and feedback interactions. Bernosconi and Konig (1999) developed a technique based on the methods of structural analysis in the field of econometrics. The basic idea is based on autoregressive modeling and quantitative measures of linear relationships between multiple time series. The concept is known as Wiener-Granger causality and the strength and direction of relationships are derived from the predictability of the models. The simplest description of the principle is that the past and present may cause the future, but the future cannot cause the past. 22

30 Multivariate time series are analyzed in the time and frequency domain for causality, but analysis is restricted to stationary responses (autoregressive modeling) and may be limited by the available amount of data (dimensionality restrictions). In addition to stationarity requirements, the method would be very ineffective in detecting instantaneous interactions. Some of the issues of stationarity can be dealt with under the assumption of piecewise stationarity and modeling each section separately. Overall, because the method can be applied in both the time and frequency domain, it can be useful to detect general cortical interactions that will help with other methods that can analyze the instantaneous interactions. Pastor et al. (2000) have also used a method for determining causal connectivity in cerebral activity using regional cerebral blood flow (RCBF) data from PET imaging studies. Their method is more suited strictly for functional imaging studies because the approach is both coarse (where regions such as the visual cortex are considered as elements) and minimalist (minimizing the number of information processors). In general, causality is better suited for long-range and regional interactions within the brain rather than local interactions between neurons. The advantage of causality over correlation is providing directionality information, but cross-correlation and the shift predictor are able to provide this information when the analysis is performed on neurons within the range of direct synaptic interactions. Nonlinear methods In almost all the methods we describe for determining correlation and connectivity of neural activity (the information-theoretic approaches being the 23

31 exceptions), the primary computation is to determine linear relationships between elements (in time or frequency between neurons or between regions). It should be expected that these methods would not detect all the possible interactions because we are dealing with elements that have several nonlinear properties. Neurons and neural networks cannot be thought of as passive linear elements because they have properties such as thresholding, intrinsic bursting, and chaos. Friston and Buchel (2000) developed a nonlinear model to analyze the feedback influences of attention in the posterier parietal cortex (PPC) on area MT responses. The model uses the Volterra series to model the nonlinear transformation and the effective connectivity is determined by solving for the unknown kernels in the convolution of the time series. The kernels are estimated by a time series expansion of temporal basis functions. Friston and Buchel (2000) apply the analysis on fmri data, but it can also be used on data with higher temporal acuity (i.e., electrophysiological recordings) by simply expanding the number of temporal basis functions. As is the case with the linear methods, the nonlinear effective connectivity is only an estimation of the possible interactions. Neural Code Theory To compound the problem of analyzing larger populations of neurons there is still much controversy over what aspects of the individual neuron s output are relevant to the neural code, or representation of information. Whether the theory is that the neurons are independent feature detectors, elements that form spatiotemporal patterns, or a feature vector, there must be an element to represent a magnitude. If it is assumed that the neural 24

32 responses are considered point processes, then this element must be some property of time. Examples of these properties are impulse rates or counts, Morse code -type patterns, precise spike arrival times, and interspike intervals (ISI). Average spike rate code Since Adrian and Zotterman (1926) discovered a relationship between the firing rate of neurons and the magnitude of sensory stimulation (touch and pressure), the rate code has been the primary property of neurons measured by neurophysiologists. In the simplest form, the firing rate is determined by listening to neural responses. More precisely, the firing rate can be measured across time by averaging responses to repeated sensory stimulations and forming the PSTH. From the PSTHs, tuning curves can be measured across feature variations to characterize a neuron for different properties of the sensory stimulation. From these tuning functions, optimal stimulus parameters and their bandwidths of the response can be determined. From these properties, the functional organization of the brain has been determined and neurons have been classified (Hubel and Weisel, 1962). It is from these tuning characteristics that Barlow (1972) formulated his cardinal cell theory. Even to this day the average firing rate is the simplest and most straightforward measurement made in neurophysiological studies. One problem with the average firing rate is that it is highly variable across stimulus repetitions (Gershon et al., 1998). This leads to the requirement of forming the averaged PST histogram across repeated stimulus presentations. The rationale was that the brain could average responses across a population instantaneously (or over a short integration time constant) in the same manner 25