Human Intracranial Recordings Research Articles

Intrinsic dynamic shapes responses to external stimulation in the human brain.

Sensory stimulation of the brain reverberates in its recurrent neuronal networks. However, current computational models of brain activity do not separate immediate sensory responses from intrinsic recurrent dynamics. We apply a vector-autoregressive model with external input (VARX), combining the concepts of "functional connectivity" and "encoding models", to intracranial recordings in humans. We find that the recurrent connectivity during rest is largely unaltered during movie watching. The intrinsic recurrent dynamic enhances and prolongs the neural responses to scene cuts, eye movements, and sounds. Failing to account for these exogenous inputs, leads to spurious connections in the intrinsic "connectivity". The model shows that an external stimulus can reduce intrinsic noise. It also shows that sensory areas have mostly outward, whereas higher-order brain areas mostly incoming connections. We conclude that the response to an external audiovisual stimulus can largely be attributed to the intrinsic dynamic of the brain, already observed during rest.

Coordinated NREM sleep oscillations among hippocampal subfields modulate synaptic plasticity in humans

The integration of hippocampal oscillations during non-rapid eye movement (NREM) sleep is crucial for memory consolidation. However, how cardinal sleep oscillations bind across various subfields of the human hippocampus to promote information transfer and synaptic plasticity remains unclear. Using human intracranial recordings from 25 epilepsy patients, we find that hippocampal subfields, including DG/CA3, CA1, and SUB, all exhibit significant delta and spindle power during NREM sleep. The DG/CA3 displays strong coupling between delta and ripple oscillations with all the other hippocampal subfields. In contrast, the regions of CA1 and SUB exhibit more precise coordination, characterized by event-level triple coupling between delta, spindle, and ripple oscillations. Furthermore, we demonstrate that the synaptic plasticity within the hippocampal circuit, as indexed by delta-wave slope, is linearly modulated by spindle power. In contrast, ripples act as a binary switch that triggers a sudden increase in delta-wave slope. Overall, these results suggest that different subfields of the hippocampus regulate one another through diverse layers of sleep oscillation synchronization, collectively facilitating information processing and synaptic plasticity during NREM sleep.

Neural populations in the language network differ in the size of their temporal receptive windows.

Despite long knowing what brain areas support language comprehension, our knowledge of the neural computations that these frontal and temporal regions implement remains limited. One important unresolved question concerns functional differences among the neural populations that comprise the language network. Here we leveraged the high spatiotemporal resolution of human intracranial recordings (n = 22) to examine responses to sentences and linguistically degraded conditions. We discovered three response profiles that differ in their temporal dynamics. These profiles appear to reflect different temporal receptive windows, with average windows of about 1, 4 and 6 words, respectively. Neural populations exhibiting these profiles are interleaved across the language network, which suggests that all language regions have direct access to distinct, multiscale representations of linguistic input-a property that may be critical for the efficiency and robustness of language processing.

Neural interactions in the human frontal cortex dissociate reward and punishment learning.

How human prefrontal and insular regions interact while maximizing rewards and minimizing punishments is unknown. Capitalizing on human intracranial recordings, we demonstrate that the functional specificity toward reward or punishment learning is better disentangled by interactions compared to local representations. Prefrontal and insular cortices display non-selective neural populations to rewards and punishments. Non-selective responses, however, give rise to context-specific interareal interactions. We identify a reward subsystem with redundant interactions between the orbitofrontal and ventromedial prefrontal cortices, with a driving role of the latter. In addition, we find a punishment subsystem with redundant interactions between the insular and dorsolateral cortices, with a driving role of the insula. Finally, switching between reward and punishment learning is mediated by synergistic interactions between the two subsystems. These results provide a unifying explanation of distributed cortical representations and interactions supporting reward and punishment learning.

Perception and Memory Reinstatement Engage Overlapping Face-Selective Regions within Human Ventral Temporal Cortex.

Humans have the remarkable ability to vividly retrieve sensory details of past events. According to the theory of sensory reinstatement, during remembering, brain regions specialized for processing specific sensory stimuli are reactivated to support content-specific retrieval. Recently, several studies have emphasized transformations in the spatial organization of these reinstated activity patterns. Specifically, studies of scene stimuli suggest a clear anterior shift in the location of retrieval activations compared with the activity observed during perception. However, it is not clear that such transformations occur universally, with inconsistent evidence for other important stimulus categories, particularly faces. One challenge in addressing this question is the careful delineation of face-selective cortices, which are interdigitated with other selective regions, in configurations that spatially differ across individuals. Therefore, we conducted a multisession neuroimaging study to first carefully map individual participants' (nine males and seven females) face-selective regions within ventral temporal cortex (VTC), followed by a second session to examine the activity patterns within these regions during face memory encoding and retrieval. While face-selective regions were expectedly engaged during face perception at encoding, memory retrieval engagement exhibited a more selective and constricted reinstatement pattern within these regions, but did not show any consistent direction of spatial transformation (e.g., anteriorization). We also report on unique human intracranial recordings from VTC under the same experimental conditions. These findings highlight the importance of considering the complex configuration of category-selective cortex in elucidating principles shaping the neural transformations that occur from perception to memory.

Theta mediated dynamics of human hippocampal-neocortical learning systems in memory formation and retrieval

Episodic memory arises as a function of dynamic interactions between the hippocampus and the neocortex, yet the mechanisms have remained elusive. Here, using human intracranial recordings during a mnemonic discrimination task, we report that 4-5 Hz (theta) power is differentially recruited during discrimination vs. overgeneralization, and its phase supports hippocampal-neocortical when memories are being formed and correctly retrieved. Interactions were largely bidirectional, with small but significant net directional biases; a hippocampus-to-neocortex bias during acquisition of new information that was subsequently correctly discriminated, and a neocortex-to-hippocampus bias during accurate discrimination of new stimuli from similar previously learned stimuli. The 4-5 Hz rhythm may facilitate the initial stages of information acquisition by neocortex during learning and the recall of stored information from cortex during retrieval. Future work should further probe these dynamics across different types of tasks and stimuli and computational models may need to be expanded accordingly to accommodate these findings.

Simulation of background neuronal activity and noise in human intracranial microwire recordings

BackgroundHuman intracranial microwire recordings allow measurement of neuronal activity in human subjects at a fine temporal and spatial scale. The recorded extracellular potentials represent a mixture of action potentials from nearby neurons, local field potentials, and other noise sources. Signal processing of these recordings is used to separate the activity of putative single neurons from other background and noise. To better understand the separation of single neuron activity, one approach is to simulate the signals produced by neurons firing action potentials combined with background activity and noise. New methodThis paper characterizes the background activity and noise in human intracranial microwire recordings and presents an accurate and efficient method of simulation using an infinite impulse response filter to color white noise. Results and comparisonThis method reproduces the power spectral density of the background activity and noise over a frequency range of 1–5000 Hz and is over 200 times faster than previously used methods. It thus facilitates large scale studies of variation of noise sources, field potentials, and processing parameters. It performs equivalently in terms of spike sorting to simulation using white noise. Another advantage is that the simulated signals are known to arise from a pseudorandom number generator and cannot be the result of detecting simulated background spiking activity. ConclusionsThis approach provides a rapid and accurate method of simulating background noise and neural activity in human intracranial microwire recordings. It is suitable for use in large scale simulations to study spike sorting in this type of recording.

Allostatic-interoceptive anticipation of social rejection

Anticipating social stress evokes strong reactions in the organism, including interoceptive modulations. However, evidence for this claim comes from behavioral studies, often with inconsistent results, and relates almost solely to the reactive and recovery phase of social stress exposure. Here, we adopted an allostatic-interoceptive predictive coding framework to study interoceptive and exteroceptive anticipatory brain responses using a social rejection task. We analyzed the heart-evoked potential (HEP) and task-related oscillatory activity of 58 adolescents via scalp EEG, and 385 human intracranial recordings of three patients with intractable epilepsy. We found that anticipatory interoceptive signals increased in the face of unexpected social outcomes, reflected in larger negative HEP modulations. Such signals emerged from key brain allostatic-interoceptive network hubs, as shown by intracranial recordings. Exteroceptive signals were characterized by early activity between 1-15 Hz across conditions, and modulated by the probabilistic anticipation of reward-related outcomes, observed over distributed brain regions. Our findings suggest that the anticipation of a social outcome is characterized by allostatic-interoceptive modulations that prepare the organism for possible rejection. These results inform our understanding of interoceptive processing and constrain neurobiological models of social stress.

Semantic novelty modulates neural responses to visual change across the human brain

Our continuous visual experience in daily life is dominated by change. Previous research has focused on visual change due to stimulus motion, eye movements or unfolding events, but not their combined impact across the brain, or their interactions with semantic novelty. We investigate the neural responses to these sources of novelty during film viewing. We analyzed intracranial recordings in humans across 6328 electrodes from 23 individuals. Responses associated with saccades and film cuts were dominant across the entire brain. Film cuts at semantic event boundaries were particularly effective in the temporal and medial temporal lobe. Saccades to visual targets with high visual novelty were also associated with strong neural responses. Specific locations in higher-order association areas showed selectivity to either high or low-novelty saccades. We conclude that neural activity associated with film cuts and eye movements is widespread across the brain and is modulated by semantic novelty.

Intracranial Electroencephalography and Deep Neural Networks Reveal Shared Substrates for Representations of Face Identity and Expressions.

According to a classical view of face perception (Bruce and Young, 1986; Haxby et al., 2000), face identity and facial expression recognition are performed by separate neural substrates (ventral and lateral temporal face-selective regions, respectively). However, recent studies challenge this view, showing that expression valence can also be decoded from ventral regions (Skerry and Saxe, 2014; Li et al., 2019), and identity from lateral regions (Anzellotti and Caramazza, 2017). These findings could be reconciled with the classical view if regions specialized for one task (either identity or expression) contain a small amount of information for the other task (that enables above-chance decoding). In this case, we would expect representations in lateral regions to be more similar to representations in deep convolutional neural networks (DCNNs) trained to recognize facial expression than to representations in DCNNs trained to recognize face identity (the converse should hold for ventral regions). We tested this hypothesis by analyzing neural responses to faces varying in identity and expression. Representational dissimilarity matrices (RDMs) computed from human intracranial recordings (n = 11 adults; 7 females) were compared with RDMs from DCNNs trained to label either identity or expression. We found that RDMs from DCNNs trained to recognize identity correlated with intracranial recordings more strongly in all regions tested-even in regions classically hypothesized to be specialized for expression. These results deviate from the classical view, suggesting that face-selective ventral and lateral regions contribute to the representation of both identity and expression.SIGNIFICANCE STATEMENT Previous work proposed that separate brain regions are specialized for the recognition of face identity and facial expression. However, identity and expression recognition mechanisms might share common brain regions instead. We tested these alternatives using deep neural networks and intracranial recordings from face-selective brain regions. Deep neural networks trained to recognize identity and networks trained to recognize expression learned representations that correlate with neural recordings. Identity-trained representations correlated with intracranial recordings more strongly in all regions tested, including regions hypothesized to be expression specialized in the classical hypothesis. These findings support the view that identity and expression recognition rely on common brain regions. This discovery may require reevaluation of the roles that the ventral and lateral neural pathways play in processing socially relevant stimuli.

208 DBS-Induced Improvement in Cognitive Control is Mediated by Theta Oscillations in Human Intracranial Recordings

INTRODUCTION: Deficits in cognitive control, defined as the ability to adjust response or attention in the face of competing or changing demands, are thought to be responsible for inflexible behaviors in neuropsychiatric diseases. Here, we seek to understand frontotemporal networks underlying dysfunctional cognitive function using sEEG recordings and deep brain stimulation (DBS) in human subjects with treatment-resistant depression (TRD). METHODS: We employed a forced two-choice cognitive control task where the strength in the target and the distractor interference of a visual stimulus was parametrically and independently varied, resulting in granular variation in response congruence. This task was performed in 13 total subjects, 10 of whom underwent sEEG implants for epilepsy monitoring, whereas 3 subjects were participating in a clinical trial of DBS for TRD and were additionally implanted with DBS leads in the ventral capsule/ventral striatum and subcallosal cingulate. RESULTS: Reaction time (RT) and accuracy worsened with increasing levels of conflict across all subjects (Cohen’s d-accuracy = 0.7, Cohen’s d-RT = 0.2) prior to stimulation. Single-trial regression analyses revealed a parametric effect of task conflict with theta-band (4-7 Hz) power across the dorsolateral PFC (dlPFC), dorsal anterior cingulate cortex, orbitofrontal cortex (OFC) as well as the insula. DBS significantly improved behavioral performance, demonstrated by reduced RT (1080 土 10 ms to 840 土 50 ms), and improved accuracy(70 土 9.8% to 90 土 5%). While we observed an overall increase in theta power across the PFC and ACC in TRD participants following DBS, the effect of conflict in theta power was reversed (theta power decreased with increasing conflict in dlPFC and OFC). CONCLUSIONS: Leveraging this unique behavioral paradigm with concurrent distributed intracranial recordings and causal manipulation broadens our understanding of neural substrates of decision-making as well as the mechanisms of therapeutic action of DBS for TRD and other psychiatric disorders.

Characterizing brain dynamics during ketamine-induced dissociation and subsequent interactions with propofol using human intracranial neurophysiology

Ketamine produces antidepressant effects in patients with treatment-resistant depression, but its usefulness is limited by its psychotropic side effects. Ketamine is thought to act via NMDA receptors and HCN1 channels to produce brain oscillations that are related to these effects. Using human intracranial recordings, we found that ketamine produces gamma oscillations in prefrontal cortex and hippocampus, structures previously implicated in ketamine’s antidepressant effects, and a 3 Hz oscillation in posteromedial cortex, previously proposed as a mechanism for its dissociative effects. We analyzed oscillatory changes after subsequent propofol administration, whose GABAergic activity antagonizes ketamine’s NMDA-mediated disinhibition, alongside a shared HCN1 inhibitory effect, to identify dynamics attributable to NMDA-mediated disinhibition versus HCN1 inhibition. Our results suggest that ketamine engages different neural circuits in distinct frequency-dependent patterns of activity to produce its antidepressant and dissociative sensory effects. These insights may help guide the development of brain dynamic biomarkers and novel therapeutics for depression.

Propofol disrupts alpha dynamics in functionally distinct thalamocortical networks during loss of consciousness

During propofol-induced general anesthesia, alpha rhythms measured using electroencephalography undergo a striking shift from posterior to anterior, termed anteriorization, where the ubiquitous waking alpha is lost and a frontal alpha emerges. The functional significance of alpha anteriorization and the precise brain regions contributing to the phenomenon are a mystery. While posterior alpha is thought to be generated by thalamocortical circuits connecting nuclei of the sensory thalamus with their cortical partners, the thalamic origins of the propofol-induced alpha remain poorly understood. Here, we used human intracranial recordings to identify regions in sensory cortices where propofol attenuates a coherent alpha network, distinct from those in the frontal cortex where it amplifies coherent alpha and beta activities. We then performed diffusion tractography between these identified regions and individual thalamic nuclei to show that the opposing dynamics of anteriorization occur within two distinct thalamocortical networks. We found that propofol disrupted a posterior alpha network structurally connected with nuclei in the sensory and sensory associational regions of the thalamus. At the same time, propofol induced a coherent alpha oscillation within prefrontal cortical areas that were connected with thalamic nuclei involved in cognition, such as the mediodorsal nucleus. The cortical and thalamic anatomy involved, as well as their known functional roles, suggests multiple means by which propofol dismantles sensory and cognitive processes to achieve loss of consciousness.

Data Archive for the BRAIN Initiative (DABI)

Data sharing is becoming ubiquitous and can be advantageous for most biomedical research. However, some data are inherently more amenable to sharing than others. For example, human intracranial neurophysiology recordings and associated multimodal data have unique features that warrant special considerations. The associated data are heterogeneous, difficult to compare, highly specific, and collected from small cohorts with treatment resistant conditions, posing additional complications when attempting to perform generalizable analyses across projects. We present the Data Archive for the BRAIN Initiative (DABI) and describe features of the platform that are designed to overcome these and other challenges. DABI is a data repository and portal for BRAIN Initiative projects that collect human and animal intracranial recordings, and it allows users to search, visualize, and analyze multimodal data from these projects. The data providers maintain full control of data sharing privileges and can organize and manage their data with a user-friendly and intuitive interface. We discuss data privacy and security concerns, example analyses from two DABI datasets, and future goals for DABI.

Frequency dependent emotion differentiation and directional coupling in amygdala, orbitofrontal and medial prefrontal cortex network with intracranial recordings

The amygdala, orbitofrontal cortex (OFC) and medial prefrontal cortex (mPFC) form a crucial part of the emotion circuit, yet their emotion induced responses and interactions have been poorly investigated with direct intracranial recordings. Such high-fidelity signals can uncover precise spectral dynamics and frequency differences in valence processing allowing novel insights on neuromodulation. Here, leveraging the unique spatio-temporal advantages of intracranial electroencephalography (iEEG) from a cohort of 35 patients with intractable epilepsy (with 71 contacts in amygdala, 31 in OFC and 43 in mPFC), we assessed the spectral dynamics and interactions between the amygdala, OFC and mPFC during an emotional picture viewing task. Task induced activity showed greater broadband gamma activity in the negative condition compared to positive condition in all the three regions. Similarly, beta activity was increased in the negative condition in the amygdala and OFC while decreased in mPFC. Furthermore, beta activity of amygdala showed significant negative association with valence ratings. Critically, model-based computational analyses revealed unidirectional connectivity from mPFC to the amygdala and bidirectional communication between OFC-amygdala and OFC-mPFC. Our findings provide direct neurophysiological evidence for a much-posited model of top-down influence of mPFC over amygdala and a bidirectional influence between OFC and the amygdala. Altogether, in a relatively large sample size with human intracranial neuronal recordings, we highlight valence-dependent spectral dynamics and dyadic coupling within the amygdala-mPFC-OFC network with implications for potential targeted neuromodulation in emotion processing.

Finger movement and coactivation predicted from intracranial brain activity using extended block-term tensor regression

Objective. We introduce extended Block-Term Tensor Regression (eBTTR), a novel regression method designed to account for the multilinear nature of human intracranial finger movement recordings. Approach. The proposed method relies on recursive Tucker decomposition combined with automatic component extraction. Main results. eBTTR outperforms state-of-the-art regression approaches, including multilinear and deep learning ones, in accurately predicting finger trajectories as well as unintentional finger coactivations. Significance. eBTTR rivals state-of-the-art approaches while being less computationally expensive which is an advantage when intracranial electrodes are implanted acutely, as part of the patient’s presurgical workup, limiting time for decoder development and testing.

Distinct population and single-neuron selectivity for executive and episodic processing in human dorsal posterior cingulate.

Posterior cingulate cortex (PCC) is an enigmatic region implicated in psychiatric and neurological disease, yet its role in cognition remains unclear. Human studies link PCC to episodic memory and default mode network (DMN), while findings from the non-human primate emphasize executive processes more associated with the cognitive control network (CCN) in humans. We hypothesized this difference reflects an important functional division between dorsal (executive) and ventral (episodic) PCC. To test this, we utilized human intracranial recordings of population and single unit activity targeting dorsal PCC during an alternated executive/episodic processing task. Dorsal PCC population responses were significantly enhanced for executive, compared to episodic, task conditions, consistent with the CCN. Single unit recordings, however, revealed four distinct functional types with unique executive (CCN) or episodic (DMN) response profiles. Our findings provide critical electrophysiological data from human PCC, bridging incongruent views within and across species, furthering our understanding of PCC function.

Cortico-cortical and thalamo-cortical connectivity during non-REM and REM sleep: Insights from intracranial recordings in humans

ObjectiveTo study changes of thalamo-cortical and cortico-cortical connectivity during wakefulness, non-Rapid Eye Movement (non-REM) sleep, including N2 and N3 stages, and REM sleep, using stereoelectroencephalography (SEEG) recording in humans. MethodsWe studied SEEG recordings of ten patients during wakefulness, non-REM sleep and REM sleep, in seven brain regions of interest including the thalamus. We calculated directed and undirected functional connectivity using a measure of non-linear correlation coefficient h2. ResultsThe thalamus was more connected to other brain regions during N2 stage and REM sleep than during N3 stage during which cortex was more connected than the thalamus. We found two significant directed links: the first from the prefrontal region to the lateral parietal region in the delta band during N3 sleep and the second from the thalamus to the insula during REM sleep. ConclusionsThese results showed that cortico-cortical connectivity is more prominent in N3 stage than in N2 and REM sleep. During REM sleep we found significant thalamo-insular connectivity, with a driving role of the thalamus. SignificanceWe found a pattern of cortical connectivity during N3 sleep concordant with antero-posterior traveling slow waves. The thalamus seemed particularly involved as a hub of connectivity during REM sleep.

Brain network dynamics codify heterogeneity in seizure evolution.

Dynamic functional brain connectivity facilitates adaptive cognition and behaviour. Abnormal alterations within such connectivity could result in disrupted functions observed across various neurological conditions. As one of the most common neurological disorders, epilepsy is defined by the seemingly random occurrence of spontaneous seizures. A central but unresolved question concerns the mechanisms by which extraordinarily diverse propagation dynamics of seizures emerge. Here, we applied a graph-theoretical approach to assess dynamic reconfigurations in the functional brain connectivity before, during and after seizures that display heterogeneous propagation patterns despite sharing similar cortical onsets. We computed time-varying functional brain connectivity networks from human intracranial recordings of 67 seizures (across 14 patients) that had a focal origin—49 of these focal seizures remained focal and 18 underwent a bilateral spread (focal to bilateral tonic-clonic seizures). We utilized functional connectivity networks estimated from interictal periods across patients as control. Our results characterize network features that quantify the underlying functional dynamics associated with the observed heterogeneity of seizure propagation across these two types of focal seizures. Decoding these network features demonstrate that bilateral propagation of seizure activity is an outcome of the imbalance of global integration and segregation in the brain prior to seizure onset. We show that there exist intrinsic network signatures preceding seizure onset that are associated with the extent to which an impending seizure will propagate throughout the brain (i.e. staying within one hemisphere versus spreading transcallosally). Additionally, these features characterize an increase in segregation and a decrease in excitability within the brain network (i.e. high modularity and low spectral radius). Importantly, seizure-type-specific differences in these features emerge several minutes prior to seizure onset, suggesting the potential utility of such measures in intervention strategies. Finally, our results reveal network characteristics after the onset that are unique to the propagation mechanisms of two most common focal seizure subtypes, indicative of distinct reconfiguration processes that may assist termination of each seizure type. Together, our findings provide insights into the relationship between the temporal evolution of seizure activity and the underlying functional connectivity dynamics. These results offer exciting avenues where graph-theoretical measures could potentially guide personalized clinical interventions for epilepsy and other neurological disorders in which extensive heterogeneity is observed across subtypes as well as across and within individual patients.

Cortical Ripples during NREM Sleep and Waking in Humans.

Hippocampal ripples index the reconstruction of spatiotemporal neuronal firing patterns essential for the consolidation of memories in the cortex during non-rapid eye movement sleep (NREM). Recently, cortical ripples in humans have been shown to enfold the replay of neuron firing patterns during cued recall. Here, using intracranial recordings from 18 patients (12 female), we show that cortical ripples also occur during NREM in humans, with similar density, oscillation frequency (∼90 Hz), duration, and amplitude to waking. Ripples occurred in all cortical regions with similar characteristics, unrelated to putative hippocampal connectivity, and were less dense and robust in higher association areas. Putative pyramidal and interneuron spiking phase-locked to cortical ripples during NREM, with phase delays consistent with ripple generation through pyramidal-interneuron feedback. Cortical ripples were smaller in amplitude than hippocampal ripples but were similar in density, frequency, and duration. Cortical ripples during NREM typically occurred just before the upstate peak, often during spindles. Upstates and spindles have previously been associated with memory consolidation, and we found that cortical ripples grouped cofiring between units within the window of spike timing-dependent plasticity. Thus, human NREM cortical ripples are as follows: ubiquitous and stereotyped with a tightly focused oscillation frequency; similar to hippocampal ripples; associated with upstates and spindles; and associated with unit cofiring. These properties are consistent with cortical ripples possibly contributing to memory consolidation and other functions during NREM in humans.SIGNIFICANCE STATEMENT In rodents, hippocampal ripples organize replay during sleep to promote memory consolidation in the cortex, where ripples also occur. However, evidence for cortical ripples in human sleep is limited, and their anatomic distribution and physiological properties are unexplored. Here, using human intracranial recordings, we demonstrate that ripples occur throughout the cortex during waking and sleep with highly stereotyped characteristics. During sleep, cortical ripples tend to occur during spindles on the down-to-upstate transition, and thus participate in a sequence of sleep waves that is important for consolidation. Furthermore, cortical ripples organize single-unit spiking with timing optimal to facilitate plasticity. Therefore, cortical ripples in humans possess essential physiological properties to support memory and other cognitive functions.