Sara Moberg

Naoya Takahashi
email: naoya.takahashi@u-bordeaux.fr

Neocortical layer 5 subclasses -from cellular properties to roles in behavior

Keywords: Neocortical layer 5, pyramidal neuron, projection type, sensory processing, rodent behavior

ABSTRACT

Layer 5 (L5) serves as the main output layer of cortical structures, where long-range projecting pyramidal neurons broadcast the columnar output to other cortical and extracortical regions of the brain. L5 pyramidal neurons are grouped into two subclasses based on their projection targets; while intratelencephalic (IT) neurons project to cortical areas and the striatum, extratelencephalic (ET) neurons project to subcortical areas such as the thalamus, midbrain, and brainstem. Each L5 subclass possesses distinct morphological and electrophysiological properties and is incorporated into a unique synaptic network. Thanks to recent advances in genetic tools and methodologies, it has now become possible to distinguish between the two subclasses in the living brain. There is increasing evidence indicating that each subclass plays a unique role in sensory processing, decision-making, and learning. This review first summarizes the anatomical and physiological properties as well as the neuromodulation of IT and ET neurons in the rodent neocortex, and then reviews recent literature on their roles in sensory processing and rodent behavior. Our ultimate goal is to provide a comprehensive understanding of the role of each subclass in cortical function by examining their operational regimes based on their cellular properties.

MAIN TEXT

The mammalian neocortex is a collection of functionally distinct circuits that are heavily interconnected and operate in parallel. Although each region of the neocortex serves different tasks, they share basic organizational rules, including a six-layer columnar structure and a diversity of cell types that comprise each layer. The main output of cortical columns is localized in layer 5 (L5), which consists of intermingled, non-overlapping populations of pyramidal cells, namely intratelencephalic (IT) and extratelencephalic (ET) neurons. L5 IT neurons project to other cortical areas and the striatum bilaterally. In contrast, ET neurons project to subcortical areas, including the ipsilateral striatum, higher-order thalamic nuclei, superior colliculus (SC), and pons. These projection patterns of IT and ET neurons can be further segregated into additional subcategories [START_REF] Tasic | Shared and distinct transcriptomic cell types across neocortical areas[END_REF][START_REF] Kim | Three Types of Cortical Layer 5 Neurons That Differ in Brain-wide Connectivity and Function[END_REF][START_REF] Economo | Distinct descending motor cortex pathways and their roles in movement[END_REF][START_REF] Rojas-Piloni | Relationships between structure, in vivo function and long-range axonal target of cortical pyramidal tract neurons[END_REF]. Here, we focus on IT and ET neurons as a whole, outlining the differences in their cellular properties and roles in sensory-motor processing and behavior. The following work mentioned in this review is mostly derived from rodents (mice and rats), unless stated otherwise.

Anatomical and morphological properties

Neocortical L5 pyramidal neurons receive inputs to their dendrites which can either be perisomatic (the basal dendrites) or very distant from the soma. These different streams of inputs are believed to serve different purposes. IT neurons have relatively small cell bodies in the upper half of L5 (L5a), whereas ET neurons with large cell bodies are found primarily in lower L5 (L5b) [START_REF] Wise | Cells of origin and terminal distribution of descending projections of the rat somatic sensory cortex[END_REF][START_REF] Kasper | Pyramidal neurons in layer 5 of the rat visual cortex. I. Correlation among cell morphology, intrinsic electrophysiological properties, and axon targets[END_REF].

One of the most prominent morphological features of L5 pyramidal neurons is the vertically extended dendritic tree, the so-called apical dendrite, which arises from the cell body and reaches the cortical surface (Fig. 1). ET neurons have a thick apical dendrite (large diameter) with many distal branches extensively arborizing within layer 1 (L1) [START_REF] Hattox | Layer V neurons in mouse cortex projecting to different targets have distinct physiological properties[END_REF][START_REF] Groh | Cell-type specific properties of pyramidal neurons in neocortex underlying a layout that is modifiable depending on the cortical area[END_REF]. Additionally, ET neurons have oblique branches protruding from the proximal trunk of the apical dendrite [START_REF] Kasper | Pyramidal neurons in layer 5 of the rat visual cortex. I. Correlation among cell morphology, intrinsic electrophysiological properties, and axon targets[END_REF][START_REF] Groh | Cell-type specific properties of pyramidal neurons in neocortex underlying a layout that is modifiable depending on the cortical area[END_REF]. In contrast, IT neurons have a thin apical dendrite (small diameter) with poorly branched distal branches and few oblique dendrites [START_REF] Kasper | Pyramidal neurons in layer 5 of the rat visual cortex. I. Correlation among cell morphology, intrinsic electrophysiological properties, and axon targets[END_REF][START_REF] Hattox | Layer V neurons in mouse cortex projecting to different targets have distinct physiological properties[END_REF][START_REF] Groh | Cell-type specific properties of pyramidal neurons in neocortex underlying a layout that is modifiable depending on the cortical area[END_REF].

Synaptic connectivity

As the neocortex consists of six layers, each of which receives different inputs from local circuits or distant brain areas, the anatomical and morphological difference between IT and ET neurons already implies differences in inputs that they primarily process. For example, L4 and L5b in the primary somatosensory cortex (S1) receive feed-forward sensory signals from the ventral posteromedial (VPM) thalamus -the primary thalamic nucleus that relays somatosensory information from the peripheries to the cortex. VPM thalamic axons terminate on basal dendrites of ET neurons, feeding ET neurons with early somatosensory information [START_REF] Constantinople | Deep cortical layers are activated directly by thalamus[END_REF]. In contrast, longrange feedback (top-down) projections from other cortical areas and higher-order thalamic nuclei are abundant in L1, where ET neurons have extensively-arborized distal branches while IT neurons have less. Notably, some feedback projections also target L5a, where they terminate at the somata and basal dendrites of IT and ET neurons [START_REF] Groh | Cell-type specific properties of pyramidal neurons in neocortex underlying a layout that is modifiable depending on the cortical area[END_REF][START_REF] Petreanu | The subcellular organization of neocortical excitatory connections[END_REF][START_REF] Mao | Long-range neuronal circuits underlying the interaction between sensory and motor cortex[END_REF]. Kim and colleagues used monosynaptic rabies tracing to compare the long-range inputs to IT and ET neurons [START_REF] Kim | Three Types of Cortical Layer 5 Neurons That Differ in Brain-wide Connectivity and Function[END_REF]. They found that ET neurons receive more input from the frontal cortical regions that are implicated in top-down modulation of brain states [START_REF] Zhang | Selective attention. Long-range and local circuits for top-down modulation of visual cortex processing[END_REF]. Notably, cortico-cortical feedback inputs preferentially target IT neurons projecting back to the source area over IT neurons projecting to other cortical regions [START_REF] Young | Laminar-specific cortico-cortical loops in mouse visual cortex[END_REF].

IT neurons thereby form looped circuits which facilitate recurrent interactions. Differences in embedded synaptic circuitry between IT and ET neurons are also evident locally within cortical columns. For each subclass, neurons are locally interconnected with other neurons in the same subclass. Interestingly, IT neurons connect to ET neurons, but ET neurons do not connect back to IT neurons [START_REF] Brown | Intracortical circuits of pyramidal neurons reflect their long-range axonal targets[END_REF][START_REF] Kiritani | Hierarchical connectivity and connection-specific dynamics in the corticospinal-corticostriatal microcircuit in mouse motor cortex[END_REF], indicating unidirectional information flow between the two neuron types.

Electrophysiological properties

Electrophysiological properties of IT and ET neurons have been extensively characterized by brain slice experiments (Fig. 1). As discussed in the previous section, IT and ET neurons can be distinguished by their distinct morphological and anatomical features. Here we included studies where authors separate neuron types based on depth and morphology, thin-tufted L5a and thicktufted L5b neurons, which are referred to as IT and ET neurons in this section, respectively.

Regarding somatic excitability and action potentials (APs), IT neurons have higher input resistance, higher firing thresholds, and higher rheobase than ET neurons [START_REF] Kasper | Pyramidal neurons in layer 5 of the rat visual cortex. I. Correlation among cell morphology, intrinsic electrophysiological properties, and axon targets[END_REF][START_REF] Hattox | Layer V neurons in mouse cortex projecting to different targets have distinct physiological properties[END_REF][START_REF] Mason | Correlations between morphology and electrophysiology of pyramidal neurons in slices of rat visual cortex. II. Electrophysiology[END_REF]. IT neurons are classified as regular spiking neurons and exhibit pronounced adaptation of the AP frequency in response to a depolarizing current pulse [START_REF] Kasper | Pyramidal neurons in layer 5 of the rat visual cortex. I. Correlation among cell morphology, intrinsic electrophysiological properties, and axon targets[END_REF][START_REF] Hattox | Layer V neurons in mouse cortex projecting to different targets have distinct physiological properties[END_REF][START_REF] Mason | Correlations between morphology and electrophysiology of pyramidal neurons in slices of rat visual cortex. II. Electrophysiology[END_REF][START_REF] Chagnac-Amitai | Burst generating and regular spiking layer 5 pyramidal neurons of rat neocortex have different morphological features[END_REF]. A large slow afterhyperpolarization (sAHP) conductance is responsible for the AP adaptation in IT neurons [START_REF] Guan | Electrophysiological properties of genetically identified subtypes of layer 5 neocortical pyramidal neurons: Ca(2)(+) dependence and differential modulation by norepinephrine[END_REF]. Individual APs of IT neurons are relatively broad in time due to a slow rate of the repolarization process [START_REF] Mason | Correlations between morphology and electrophysiology of pyramidal neurons in slices of rat visual cortex. II. Electrophysiology[END_REF].

I n contrast, ET neurons are distinguished by the tendency for their APs to appear in highfrequency bursts -intrinsically bursting [START_REF] Kasper | Pyramidal neurons in layer 5 of the rat visual cortex. I. Correlation among cell morphology, intrinsic electrophysiological properties, and axon targets[END_REF][START_REF] Hattox | Layer V neurons in mouse cortex projecting to different targets have distinct physiological properties[END_REF][START_REF] Mason | Correlations between morphology and electrophysiology of pyramidal neurons in slices of rat visual cortex. II. Electrophysiology[END_REF][START_REF] Chagnac-Amitai | Burst generating and regular spiking layer 5 pyramidal neurons of rat neocortex have different morphological features[END_REF]. In response to a current pulse, ET neurons respond with a brief burst (often seen as an AP doublet) followed by repetitive single non-adapting APs. ET neurons have very little sAHPs, which may explain their non-adapting firing pattern [START_REF] Mason | Correlations between morphology and electrophysiology of pyramidal neurons in slices of rat visual cortex. II. Electrophysiology[END_REF][START_REF] Guan | Electrophysiological properties of genetically identified subtypes of layer 5 neocortical pyramidal neurons: Ca(2)(+) dependence and differential modulation by norepinephrine[END_REF]. Individual APs of ET neurons are narrower and accompanied by more significant fast and medium AHPs (fAHPs and mAHPs, respectively) and afterdepolarizations (ADPs) than those of IT neurons [START_REF] Hattox | Layer V neurons in mouse cortex projecting to different targets have distinct physiological properties[END_REF]. The difference in polarization and repolarization mechanisms between the two subclasses may also impact the firing modes, explaining why ET neurons are more prone to burst firing.

Another notable difference between the two subclasses is a hyperpolarization-activated cation current (Ih), measured as a "sag" potential in response to hyperpolarization. ET neurons show a pronounced sag response to hyperpolarizing currents, whereas IT neurons show little sag response [START_REF] Kasper | Pyramidal neurons in layer 5 of the rat visual cortex. I. Correlation among cell morphology, intrinsic electrophysiological properties, and axon targets[END_REF][START_REF] Hattox | Layer V neurons in mouse cortex projecting to different targets have distinct physiological properties[END_REF][START_REF] Mason | Correlations between morphology and electrophysiology of pyramidal neurons in slices of rat visual cortex. II. Electrophysiology[END_REF]. Ih currents are mediated by hyperpolarization-activated cyclic nucleotide-gated nonselective cation (HCN) channels and prevent the membrane potential from being too hyperpolarized. HCN channels are tonically active near the resting membrane potential, and thus act as a leak conductance that impedes the accumulation of synaptic potentials [START_REF] Biel | Hyperpolarization-activated cation channels: from genes to function[END_REF]. Interestingly, HCN channels are expressed in a gradient along the somatodendritic axis of ET neurons and are more abundant in distal dendrites [START_REF] Berger | High I(h) channel density in the distal apical dendrite of layer V pyramidal cells increases bidirectional attenuation of EPSPs[END_REF][START_REF] Lorincz | Polarized and compartmentdependent distribution of HCN1 in pyramidal cell dendrites[END_REF][START_REF] Kole | Single Ih channels in pyramidal neuron dendrites: properties, distribution, and impact on action potential output[END_REF][START_REF] Harnett | Distribution and function of HCN channels in the apical dendritic tuft of neocortical pyramidal neurons[END_REF]. The impact of Ih conductance on dendritic excitability and synaptic integration is discussed in the next section.

Synaptic integration in dendrites

Both IT and ET neurons in L5 have characteristic dendrites, which electrically bridge between the distal dendrites and somata and play an essential role in regulating neuronal excitability. Here we discuss the differences in the integration of synaptic inputs at the apical dendrites between IT and ET neurons. The classical view on synaptic integration in a neuron is that synaptic potentials from input sites passively propagate along dendritic arbors to the AP initiation site -axon initial segment [START_REF] Rall | Branching dendritic trees and motoneuron membrane resistivity[END_REF][START_REF] Rall | Distinguishing theoretical synaptic potentials computed for different somadendritic distributions of synaptic input[END_REF]. The amplitude of a synaptic potential declines exponentially with travel distance and branching points. The long, vertically elongated apical dendrites of L5 pyramidal neurons therefore function as an electrical filter that attenuates distal inputs and prevents them from affecting somatic APs. By simultaneous whole-cell recording of the soma and apical dendrite of single L5 neurons in slices, Williams and Stuart demonstrated that synaptic potentials at the distal dendrites are attenuated approximately 40-fold by the time they reach the soma [START_REF] Williams | Dependence of EPSP efficacy on synapse location in neocortical pyramidal neurons[END_REF]. The attenuation is even more severe in more realistic in vivo conditions, where dendrites are bombarded with spontaneous synaptic inputs that shunt out current [START_REF] Bernander | Synaptic background activity influences spatiotemporal integration in single pyramidal cells[END_REF][START_REF] Williams | Spatial compartmentalization and functional impact of conductance in pyramidal neurons[END_REF].

Additionally, Ih conductances spur attenuation of synaptic potentials in dendrites. In ET neurons, due to the dendritic gradient of the HCN channel density, Ih conductances severely impact the summation and propagation of synaptic potentials in the distal apical dendrites [START_REF] Berger | High I(h) channel density in the distal apical dendrite of layer V pyramidal cells increases bidirectional attenuation of EPSPs[END_REF][START_REF] Harnett | Distribution and function of HCN channels in the apical dendritic tuft of neocortical pyramidal neurons[END_REF]. The differences in morphology and HCN channel expression between IT and ET dendrites imply their distinct filtering properties for processing distal inputs. In the medial prefrontal cortex (mPFC), Dembow and colleagues have indeed shown that the apical dendrites of IT and ET neurons exhibit distinct subthreshold voltage responses to identical synaptic inputs [START_REF] Dembrow | Temporal dynamics of L5 dendrites in medial prefrontal cortex regulate integration versus coincidence detection of afferent inputs[END_REF]. They found that the amplitude of synaptic potentials attenuated similarly between the subtypes, but the time courses of the transferred potentials were significantly stretched in IT neurons. The difference in the time window for input summation suggests that ET dendrites act more as coincidence detectors while IT dendrites act as temporal integrators. Nevertheless, transferred synaptic potentials from distal dendrites are still very small at the soma, failing to elicit APs. Small voltage fluctuations in distal dendrites are therefore electrically segregated from the soma in both IT and ET neurons.

Active dendritic properties

The distal dendrites in L1 are innervated extensively by feedback projections from various cortical areas and higher-order thalamic regions. How can these inputs influence the somatic output, given the strong attenuation they undergo? Dendrites not only passively integrate or transmit electrical signals, but can also actively process incoming inputs and transform them into AP outputs. The regenerative activation of voltage-gated ion channels, which are distributed throughout the dendritic membrane, can lead to the generation of spikes in dendrites, so-called dendritic spikes.

For example, somatically-generated APs can propagate back into the apical dendrite with little attenuation through activation of dendritic sodium channels along the apical trunk, so-called backpropagating APs (bAPs) [START_REF] Stuart | Active propagation of somatic action potentials into neocortical pyramidal cell dendrites[END_REF][START_REF] Stuart | Action potential initiation and propagation in rat neocortical pyramidal neurons[END_REF]. bAPs provide a rapid retrograde signal that could modulate membrane excitability of the distal dendrites and interact with synaptic inputs to regulate dendritic plasticity. Although bAPs are seen in both IT and ET neurons, their propagation appears less efficient in IT neurons, probably due to a lower density of active conductances in the apical dendrite [START_REF] Grewe | Back-Propagation of Physiological Action Potential Output in Dendrites of Slender-Tufted L5A Pyramidal Neurons[END_REF]. In addition to voltage-sensitive sodium channels, apical dendrites also express various types of voltage-sensitive calcium channels, including both low-and high-threshold channels [START_REF] Perez-Garci | Inhibition of dendritic Ca2+ spikes by GABAB receptors in cortical pyramidal neurons is mediated by a direct Gi/o-beta-subunit interaction with Cav1 channels[END_REF]. The regenerative activation of these channels causes long-lasting plateau potentials, so-called dendritic calcium spikes [START_REF] Amitai | Regenerative activity in apical dendrites of pyramidal cells in neocortex[END_REF][START_REF] Yuste | Ca2+ accumulations in dendrites of neocortical pyramidal neurons: an apical band and evidence for two functional compartments[END_REF][START_REF] Schiller | Calcium action potentials restricted to distal apical dendrites of rat neocortical pyramidal neurons[END_REF][START_REF] Larkum | A new cellular mechanism for coupling inputs arriving at different cortical layers[END_REF]. Calcium spikes are generated at the apical trunk regions and propagate forward to the soma with little attenuation and strongly depolarize the somatic membrane, leading to bursts of high-frequency APs [START_REF] Larkum | A new cellular mechanism for coupling inputs arriving at different cortical layers[END_REF][START_REF] Williams | Mechanisms and consequences of action potential burst firing in rat neocortical pyramidal neurons[END_REF]. Synchronized inputs to distal dendrites or a train of high-frequency bAPs can trigger calcium spikes [START_REF] Schiller | Calcium action potentials restricted to distal apical dendrites of rat neocortical pyramidal neurons[END_REF][START_REF] Larkum | Calcium electrogenesis in distal apical dendrites of layer 5 pyramidal cells at a critical frequency of back-propagating action potentials[END_REF]. Larkum and colleagues have demonstrated that distal inputs coincided with bAPs efficiently elicit calcium spikes, which are therefore suggested as coincidence detectors for correlating feedback inputs arriving at the distal dendrites with feed-forward sensory inputs at the soma [START_REF] Larkum | A new cellular mechanism for coupling inputs arriving at different cortical layers[END_REF]. Responding to varying distal inputs, calcium spikes also serve as a mechanism for modulating the gain of the somatic AP output by changing the firing mode from isolated spikes to bursting [START_REF] Larkum | Top-down dendritic input increases the gain of layer 5 pyramidal neurons[END_REF]. Thus, calcium spikes are essential for integrating distal input with somatic activity. It is noteworthy that most studies on calcium spikes have been conducted on ET neurons. We still know very little about calcium spikes in IT neurons, but recent studies indicate that they might have less active calcium conductances or have a higher threshold for the generation of calcium spikes compared to those in ET neurons [START_REF] Takahashi | Active dendritic currents gate descending cortical outputs in perception[END_REF][START_REF] Grewe | Back-Propagation of Physiological Action Potential Output in Dendrites of Slender-Tufted L5A Pyramidal Neurons[END_REF]. Dembow and colleagues measured the dendritic and somatic voltage responses of IT and ET neurons in mPFC to depolarizing currents injected into the apical dendrites [START_REF] Dembrow | Temporal dynamics of L5 dendrites in medial prefrontal cortex regulate integration versus coincidence detection of afferent inputs[END_REF].

Interestingly, while isolated distal synaptic inputs fail to propagate to the soma, the strong depolarizing currents reliably elicited calcium spikes in the apical dendrites of ET neurons, which drove burst firing at the soma. In contrast, in IT neurons, the injected currents failed to generate calcium spikes but directly triggered somatic APs.

Neuromodulation

The differences between IT and ET neurons include their response to neuromodulatory inputs.

Neuromodulatory projections typically extend throughout the neocortex, but most work on the influence of neuromodulators has concentrated on the mPFC circuit due to its involvement in higher-order cognitive functions. We therefore present work mainly from mPFC, if not stated otherwise, but it is important to keep in mind that neuromodulatory effects may differ between brain regions. Here, we focus on dopamine (DA), acetylcholine (ACh), norepinephrine (NE), and serotonin (5-HT) as these neuromodulators have shown to exert distinct effects on IT and ET neurons due to differential expression of each respective receptor subunit and subtypes.

Dopamine

It has been proposed that DA projections from the ventral tegmental area may predominantly target cortical pyramidal neurons in L5 due to their innervation patterns in the lower layers of the cortex [START_REF] Descarries | Regional and laminar density of the dopamine innervation in adult rat cerebral cortex[END_REF][START_REF] Nomura | Noradrenalin and dopamine receptors both control cAMP-PKA signaling throughout the cerebral cortex[END_REF]. DA receptors can be segregated into two kinds of G-protein-coupled receptors: D1receptor and D2-receptor. Differential expressions of DA receptors are also found in IT and ET neurons in PFC [START_REF] Seong | D1 receptor modulation of action potential firing in a subpopulation of layer 5 pyramidal neurons in the prefrontal cortex[END_REF]. D1-receptors were found to be predominantly expressed in L5 neurons with little Ih current, putative IT neurons, but activation of these receptors only slightly enhanced their excitability [START_REF] Xing | Distinct roles for prefrontal dopamine D1 and D2 neurons in social hierarchy[END_REF]. D2-receptors, however, were found predominantly on ET neurons [START_REF] Gee | Synaptic Activity Unmasks Dopamine D2 Receptor Modulation of a Specific Class of Layer V Pyramidal Neurons in Prefrontal Cortex[END_REF] and activation of D2-receptors on these neurons drives calcium channel-dependent ADPs, leading to sustained repetitive firing. How this selective enhancement of excitability in IT and ET neurons upon DA release in the cortex relates to separate functional pathways remains unknown.

Acetylcholine

In contrast to dopaminergic inputs, cholinergic fibers from the basal forebrain innervate the cortex in both L1 and L6 (47) and there's a plethora of evidence suggesting differential responses in IT and ET neurons and a central role of ACh in cognitive functions such as arousal and attention [START_REF] Letzkus | Disinhibition, a Circuit Mechanism for Associative Learning and Memory[END_REF][START_REF] Froemke | A synaptic memory trace for cortical receptive field plasticity[END_REF][START_REF] Nunez | Cholinergic-mediated response enhancement in barrel cortex layer V pyramidal neurons[END_REF][START_REF] Herrero | Muscarinic and Nicotinic Contribution to Contrast Sensitivity of Macaque Area V1 Neurons[END_REF][START_REF] Baker | Preferential cholinergic excitation of corticopontine neurons[END_REF]. Overall, IT neurons exhibit only a slight enhancement in firing in response to ACh compared to ET neurons that shows a robust persistence in firing [START_REF] Baker | Preferential cholinergic excitation of corticopontine neurons[END_REF]. ACh acts through two different kinds of receptors, the slower G-protein-coupled muscarinic receptors (mAChR) and the faster ionotropic nicotinic receptors (nAChR). mAChRs express on both IT and ET neurons and exert somewhat similar effects on each type; an initial transient hyperpolarization followed by a prolonged depolarization [START_REF] Baker | Preferential cholinergic excitation of corticopontine neurons[END_REF]. However, the mAChR-mediated inhibition is more pronounced in IT neurons, while excitatory responses are greater and longer lasting in ET neurons, which can lead to persistent firing [START_REF] Joshi | Cell-Specific Cholinergic Modulation of Excitability of Layer 5B Principal Neurons in Mouse Auditory Cortex[END_REF]. Recent work has shown that local ACh release enhances calcium spikes in the apical dendrites of ET neurons via activation of R-type calcium channels mediated by mAChRs [START_REF] Williams | A Dendritic Substrate for the Cholinergic Control of Neocortical Output Neurons[END_REF], converting ET neurons from regular to burst firing mode [START_REF] Nunez | Cholinergic-mediated response enhancement in barrel cortex layer V pyramidal neurons[END_REF]. Moreover, Suzuki and Larkum revealed that mAChRs are necessary for dendro-somatic coupling in L5 pyramidal neurons in S1 [START_REF] Suzuki | General Anesthesia Decouples Cortical Pyramidal Neurons[END_REF]. Overall, transient ACh release preferentially promotes ET neuronal firing over IT neurons via mAChR activation. nAChRs consist of a combination of α and β subunits. The predominant nAChRs found in sensory cortices are hetero-pentameric receptors composed of α4 and β2 subunits and homo-pentameric receptors composed of α7 subunits [START_REF] Gotti | Structural and functional diversity of native brain neuronal nicotinic receptors[END_REF]. In the rodent neocortex, local ACh application or activation of cholinergic fibers evokes nAChR-mediated, transient excitatory currents in L5 neurons [START_REF] Zolles | Functional expression of nicotinic acetylcholine receptors in rat neocortical layer 5 pyramidal cells[END_REF], in both IT and ET neurons [START_REF] Joshi | Cell-Specific Cholinergic Modulation of Excitability of Layer 5B Principal Neurons in Mouse Auditory Cortex[END_REF]. Interestingly, L5b (putative ET) neurons in the rat primary auditory cortex (A1) exhibit an age-related decrease in ACh sensitivity due to a shift in the nAChR subunit composition with an increase in the α7 subunit [START_REF] Ghimire | Nicotinic Receptor Subunit Distribution in Auditory Cortex: Impact of Aging on Receptor Number and Function[END_REF], which has lower ACh affinity [START_REF] Fenster | Influence of Subunit Composition on Desensitization of Neuronal Acetylcholine Receptors at Low Concentrations of Nicotine[END_REF].

Norepinephrine

NE is released from the locus coeruleus (LC), which projects to many areas, including the neocortex [START_REF] Nomura | Noradrenalin and dopamine receptors both control cAMP-PKA signaling throughout the cerebral cortex[END_REF][START_REF] Loughlin | Efferent projections of nucleus locus coeruleus: Morphologic subpopulations have different efferent targets[END_REF], and has an essential role in sensory processing and arousal [START_REF] Sara | The locus coeruleus and noradrenergic modulation of cognition[END_REF][START_REF] Bouret | Locus coeruleus activation modulates firing rate and temporal organization of odour-induced single-cell responses in rat piriform cortex[END_REF][START_REF] Carter | Tuning arousal with optogenetic modulation of locus coeruleus neurons[END_REF][START_REF] Janitzky | Optogenetic silencing of locus coeruleus activity in mice impairs cognitive flexibility in an attentional set-shifting task[END_REF][START_REF] Hayat | Locus coeruleus norepinephrine activity mediates sensory-evoked awakenings from sleep[END_REF]. In contrast to DA neurons, NE neurons generally spread across all layers of the cortex [START_REF] Nomura | Noradrenalin and dopamine receptors both control cAMP-PKA signaling throughout the cerebral cortex[END_REF], providing a broader blanket of modulation. Somewhat similar effects of NE have been observed in IT and ET neurons, where IT neurons showed a slight increase in AP firing and ET neurons exhibit a greater increase [START_REF] Dembrow | Projection-specific neuromodulation of medial prefrontal cortex neurons[END_REF]. Activation of α2A adrenergic receptors strengthens working memory through inactivating cAMP, leading to the closure of HCN channels in rodents and monkeys [START_REF] Franowicz | Mutation of the α2A-Adrenoceptor Impairs Working Memory Performance and Annuls Cognitive Enhancement by Guanfacine[END_REF][START_REF] Wang | Alpha2Aadrenoceptors strengthen working memory networks by inhibiting cAMP-HCN channel signaling in prefrontal cortex[END_REF].

Moreover, this was found to dominantly affect the apical dendrites of L5 pyramidal neurons in PFC, where α2A adrenergic receptor activation reduces the threshold for which bAPs can trigger dendritic calcium spikes leading to increased dendritic excitability [START_REF] Barth | α2-adrenergic receptors modify dendritic spike generation via HCN channels in the prefrontal cortex[END_REF][START_REF] Labarrera | Adrenergic Modulation Regulates the Dendritic Excitability of Layer 5 Pyramidal Neurons In Vivo[END_REF]. Using two-photon calcium imaging in vivo, Labarerra and colleagues showed that activation of α2A adrenergic receptors increases the dendro-somatic coupling through blockage of Ih [START_REF] Labarrera | Adrenergic Modulation Regulates the Dendritic Excitability of Layer 5 Pyramidal Neurons In Vivo[END_REF]. Blockage of α2A adrenergic receptors increased the duration and reduced the sag amplitude, similarly to the abovementioned dynamics of ET neurons (this was however not specified in these studies).

5-HT

5-HT afferents arrive in the neocortex mainly from the dorsal raphe nucleus [START_REF] Bang | Projections and interconnections of genetically defined serotonin neurons in mice[END_REF]. The effect exerted on different neuron types differs depending on the composition of receptor subtypes on the postsynaptic neuron. 5-HT receptors are classified into seven main groups, but L5 pyramidal neurons mainly express a combination of 5-HT1A and 5-HT2A receptors [START_REF] Amargós-Bosch | Coexpression and In Vivo Interaction of Serotonin1A and Serotonin2A Receptors in Pyramidal Neurons of Prefrontal Cortex[END_REF]. Generally, activation of 5-HT1A receptors has an inhibitory effect, while 5-HT2A receptor activation has an excitatory effect. 5-HT receptors on ET neurons in mice consist of predominantly inhibitory 5-HT1A receptors, and application of 5-HT to mPFC ET neurons leads to an increase of Ih currents (73-75). 5-HT could therefore have an opposite effect on the excitability of ET neurons compared to NE, indicating a potential competitive mechanism to regulate attention and arousal in different contexts.

IT neurons, however, consist of a combination of 5-HT1A and 5-HT2A receptors and 5-HT application typically causes depolarization [START_REF] Avesar | Selective serotonergic excitation of callosal projection neurons[END_REF][START_REF] Stephens | Activity-dependent serotonergic excitation of callosal projection neurons in the mouse prefrontal cortex[END_REF][START_REF] Elliott | Serotonin Differentially Regulates L5 Pyramidal Cell Classes of the Medial Prefrontal Cortex in Rats and Mice[END_REF]. This can either be direct or biphasic following a brief inhibitory 5-HT1A activation followed by the excitatory 5-HT2A effect.

Overall, the neuromodulatory effects of 5-HT are strikingly different in IT and ET neurons regardless of brain regions, although inconsistencies exist between cell types and species [START_REF] Elliott | Serotonin Differentially Regulates L5 Pyramidal Cell Classes of the Medial Prefrontal Cortex in Rats and Mice[END_REF].

Further work is needed to elucidate how these differences relate to the IT-ET interplay during different contexts and conditions.

Computational roles of IT and ET neurons

We have so far discussed the morphological, anatomical, and physiological properties of IT and ET neurons in neocortical L5, as well as the neuromodulatory influence on these subtypes. It is evident that in each one of these aspects, there are clear differences between IT and ET neurons.

The computational roles of IT versus ET neurons have been discussed by Harris and Shephard [START_REF] Harris | The neocortical circuit: themes and variations[END_REF]. They suggest that ET neurons act as downstream elements in the local circuit, which integrates the results of local computations with direct thalamic inputs and efficiently broadcasts the results to distant subcortical structures. In contrast, local and long-range connectivity of IT neurons forms the backbone of communication within and between cortical areas and hemispheres.

Their outputs go to other cortical areas and striatum, as well as locally to ET neurons. Here, we build on these insights and provide our perspective on the computational role of IT and ET neurons in sensory processing by taking into account the differences in their physiological and anatomical properties (Fig. 2).

IT neurons form a dense recurrent network through their local and long-range connections. In such network configurations, activity patterns converge to stable states (fixed points) in activity space (repertoire of possible activity patterns), known as attractor dynamics. Attractor networks have been proposed as a model of cortical learning and memory [START_REF] Hopfield | Neural networks and physical systems with emergent collective computational abilities[END_REF][START_REF] Amit | Model of global spontaneous activity and local structured activity during delay periods in the cerebral cortex[END_REF]. The synaptic connectivity between neurons determines the activity space and its multiple attractor points. Each attractor state thus represents a stored memory, which reflects an internal model held within the cortex -inner belief of how the world should be or representation of behavior context and task rules [START_REF] Yang | Task representations in neural networks trained to perform many cognitive tasks[END_REF]. Based on available sensory data, attractor networks construct probabilistically plausible internal models.

The IT neuronal network is well suited for building such internal models based on experience and learning. Interestingly, a computational study indicates that the spike frequency adaptation, an electrophysiological characteristic of IT neurons, can allow the network to integrate temporally dispersed information seamlessly into ongoing network activity [START_REF] Salaj | Spike frequency adaptation supports network computations on temporally dispersed information[END_REF]. In sensory information processing, internal models are essential for interpreting incoming sensory input, such as detecting deviations from expectations, filling in missing information, and assigning weights to information relevant to tasks and contexts [START_REF] Gilbert | Brain states: top-down influences in sensory processing[END_REF]. Neuromodulators such as 5-HT regulate the intrinsic excitability and synaptic integration of IT neurons. Such neuromodulatory influence can enhance the capability and flexibility of computation in the IT neuronal network, dynamically adapting the internal model to diverse behavioral demands [START_REF] Tsuda | Neuromodulators generate multiple context-relevant behaviors in a recurrent neural network by shifting activity hypertubes[END_REF]. Lastly, but perhaps most importantly, internal models generated by the IT neurons are fed back to ET neurons.

ET neurons receive broadly-tuned sensory information mainly at the basal dendrites, while nonsensory (feedback) information, mostly from IT neurons in distant cortical areas, targets the distal apical dendrites. Due to the electrical distance between the distal dendrites and the soma, feedback inputs alone have little influence on the somatic AP output. Feedback inputs act as a filter that gates coincidental sensory inputs arriving at the soma through activating calcium spikes in the apical dendrites. The spatial and temporal characteristics of the filter depend on the target specificity of the feedback inputs and their duration, respectively. The spatiotemporal characteristics are further shaped by the intrinsic properties of the dendritic membrane. Finally, gated sensory information is boosted into bursts of APs and transmitted to the subcortical regions.

It is intriguing that neuromodulators such as ACh and NE, which are elevated in arousal and alert states, promote calcium spikes in ET neurons, thereby enhancing the interaction between feedback and sensory information. As a result, the internal model could edit or highlight specific cortical representations of the sensory environment and output the results to subcortical structures to exert control over subcortical sensory-motor processing.

Interestingly, due to the lack of connection from ET to IT neurons, the gated ET output does not directly influence the activity of local IT neurons. Thus, internal models are retained in IT neurons without being affected at a local scale. Although there is no direct interference from ET to IT neuronal activity, the ET output to subcortical regions could influence the activity of IT neurons via subcortical-to-cortical ascending pathways. One such pathway is thalamocortical projections via the higher-order thalamus [START_REF] Sherman | Thalamus plays a central role in ongoing cortical functioning[END_REF], which have diffusive innervation in the cortex. Thus, in principle, the output of gated ET neurons can interfere with the activity of IT neurons, not locally but rather on a global scale, which may promote a transition in attractor states in IT neuronal activity.

In summary, these two subclasses of L5 pyramidal neurons are likely to play distinct roles in cortical computation, leading to the natural hypothesis that they are involved in different brain functions. Next, we discuss what role each subclass of L5 pyramidal neurons plays in sensory processing and behavior.

Sensory processing

The following body of work mainly separates ET and IT neurons depending on the depth of the somata. Here, we generally consider the neurons in L5b to be the ET neurons and neurons in L5a to be IT neurons. ET neurons in L5b have slightly more spontaneous activity than IT neurons in L5a [START_REF] Manns | Sub-and suprathreshold receptive field properties of pyramidal neurones in layers 5A and 5B of rat somatosensory barrel cortex[END_REF].

Differences in sensory receptive fields of IT and ET neurons have been well studied. For instance, IT neurons seem to have a narrower receptive field to passive stimulus and a rapid onset of response in contrast to ET neuronal responses which have broader receptive fields and slow onset of response [START_REF] Manns | Sub-and suprathreshold receptive field properties of pyramidal neurones in layers 5A and 5B of rat somatosensory barrel cortex[END_REF][START_REF] Palmer | Visual receptive fields of single striate corical units projecting to the superior colliculus in the cat[END_REF][START_REF] Finlay | Quantitative studies of single-cell properties in monkey striate cortex. IV. Corticotectal cells[END_REF]. During whisking without object contact, IT neurons in S1 significantly increase their firing compared to ET neurons [START_REF] De Kock | Spiking in primary somatosensory cortex during natural whisking in awake head-restrained rats is cell-type specific[END_REF]. In contrast, during active whisker touch, a subset of ET neurons goes into burst firing (but not IT neurons), which could reliably infer the touch event [START_REF] De Kock | High-frequency burst spiking in layer 5 thick-tufted pyramids of rat primary somatosensory cortex encodes exploratory touch[END_REF]. Burst firing of ET neurons could therefore convey relevant touch information of the surrounding environment to subcortical targets. The reasons for these separate sensory response properties and how they relate to animal behavior will be discussed in the next section.

Behavior

In recent years, there has been a surge of evidence of the distinct functional properties of IT and ET neurons in cognitive behavior [START_REF] Takahashi | Active dendritic currents gate descending cortical outputs in perception[END_REF][START_REF] Li | A motor cortex circuit for motor planning and movement[END_REF][START_REF] Bae | Parallel processing of working memory and temporal information by distinct types of cortical projection neurons[END_REF][START_REF] Currie | Spatiotemporal organization of movement-invariant and movement-specific signaling in the output layer of motor cortex[END_REF][START_REF] Musall | Pyramidal cell types drive functionally distinct cortical activity patterns during decision-making[END_REF][START_REF] Heindorf | Reduction of layer 5 mediated long-range cortical communication by antipsychotic drugs[END_REF][START_REF] Park | Motor cortical output for skilled forelimb movement is selectively distributed across projection neuron classes[END_REF][START_REF] Mohan | Cortical glutamatergic projection neuron types contribute to distinct functional subnetworks[END_REF]. With recent development of combining whole-brain imaging with newly developed mouse lines, it is now possible to record brain-wide activity of IT neurons (including both L5 and L2/3 IT neurons) and L5 ET neurons. It seems that IT neurons are strongly engaged in preparatory stages of various tasks as they are more active in sensory regions during sensory sampling phases [START_REF] Musall | Pyramidal cell types drive functionally distinct cortical activity patterns during decision-making[END_REF][START_REF] Mohan | Cortical glutamatergic projection neuron types contribute to distinct functional subnetworks[END_REF]. This trend is consistent in other areas such as mPFC where they retain goal specific responses prior to action initiation [START_REF] Bae | Parallel processing of working memory and temporal information by distinct types of cortical projection neurons[END_REF] and preparatory movement activities in motor structures [START_REF] Currie | Spatiotemporal organization of movement-invariant and movement-specific signaling in the output layer of motor cortex[END_REF][START_REF] Heindorf | Reduction of layer 5 mediated long-range cortical communication by antipsychotic drugs[END_REF]. Inhibiting the IT neurons during sampling periods only weakly affects the behavioral outcome, suggesting a minor role in regulating behavioral output.

ET neurons are instead engaged at later stages such as decision phases and during action initiation [START_REF] Bae | Parallel processing of working memory and temporal information by distinct types of cortical projection neurons[END_REF][START_REF] Currie | Spatiotemporal organization of movement-invariant and movement-specific signaling in the output layer of motor cortex[END_REF][START_REF] Musall | Pyramidal cell types drive functionally distinct cortical activity patterns during decision-making[END_REF][START_REF] Mohan | Cortical glutamatergic projection neuron types contribute to distinct functional subnetworks[END_REF]. They have been found to be more prominent in motor structures where they drive movement initiation [START_REF] Musall | Pyramidal cell types drive functionally distinct cortical activity patterns during decision-making[END_REF][START_REF] Mohan | Cortical glutamatergic projection neuron types contribute to distinct functional subnetworks[END_REF]. However, ET neuronal activity can also be seen in sensory areas where they regulate detection of behaviorally-relevant sensory stimuli [START_REF] Takahashi | Active dendritic currents gate descending cortical outputs in perception[END_REF][START_REF] Ruediger | Learning speed and detection sensitivity controlled by distinct cortico-fugal neurons in visual cortex[END_REF]. Thus, this reinforces the concept of IT neurons maintaining and updating cortical neurons for more efficient computations and ET neurons broadcasting this updated information to the subcortical structures.

With the recent advancements of modern tools, we have come a long way to perturb IT and ET neuronal circuits. There is a barrage of work in recent years showing specific roles of distinct circuits of IT and ET neurons but efforts to delineate the multiple circuits of IT and ET neurons and compare those across each other remains extremely difficult. As the tools and technology advances, so does our ability to probe these circuits and our understanding on how they relate to each other. Although evidence of separate L5 IT neuronal circuits in behavior remains limited, more work has been done on ET neurons and their specific downstream targets, presenting rich segregation of roles during behavior. Next, we elaborate further on these distinct differences in target-selective ET neurons.

Superior Colliculus (SC)

SC-targeting ET neurons are more sensitive to contrast thresholds, thus modulating the sensitivity to detect relevant sensory stimuli [START_REF] Takahashi | Active dendritic currents gate descending cortical outputs in perception[END_REF][START_REF] Ruediger | Learning speed and detection sensitivity controlled by distinct cortico-fugal neurons in visual cortex[END_REF][START_REF] Lur | Projection-Specific Visual Feature Encoding by Layer 5 Cortical Subnetworks[END_REF]. Using two-photon imaging, Takahashi and colleagues recorded the activity from the apical dendrites in L1 of ET neurons [START_REF] Takahashi | Active dendritic currents gate descending cortical outputs in perception[END_REF]. They showed that dendritic calcium spikes dynamically regulate sensory responses in ET neurons in S1 for tactile detection in a context-dependent manner. Local chemogenetic inactivation of the ET neuronal output in SC shifted the mouse's perceptual threshold towards higher intensities, indicating the importance of the corticocollicular pathway in detecting sensory stimuli relevant to a given context. SC is an excellent structure for integrating multiple sensory modalities [START_REF] Zingg | AAV-Mediated Anterograde Transsynaptic Tagging: Mapping Corticocollicular Input-Defined Neural Pathways for Defense Behaviors[END_REF][START_REF] May | The mammalian superior colliculus: laminar structure and connections[END_REF] from both cortical and peripheral sources via the brainstem (100-102). SCs could therefore function as a multisensory integrator for guiding behavior by comparing the contextually important signals transmitted by ET neurons with the incoming peripheral sensory inputs.

Pons

The pons receives extensive cortical input from ET neurons and projects selectively to the cerebellum, serving as an essential node for cortico-cerebellar communication. Perturbation of the ponto-cerebellar pathway does not affect movement initiation, but disrupts the success rate and precision of the movement (103). Using mice trained for a variable forelimb movement task with a joystick, Park and colleagues showed that pons-projecting ET neurons in the primary motor cortex (M1) represent movement direction [START_REF] Park | Motor cortical output for skilled forelimb movement is selectively distributed across projection neuron classes[END_REF]. Interestingly, in the same behavior, striatumprojecting IT neuronal activity in M1 was preferentially tuned to movement amplitude, with little to movement direction. In the primary visual cortex (V1), pons-projecting ET neurons were found to be engaged in a visually-cued conditioned eyeblink task (104), indicating a role for the corticoponto-cerebellar pathway in reflexive motor responses (105). In contrast, a similar sensory detection task, but requiring goal-directed, non-directional movements, does not require the output of ET neurons in the sensory cortex to pons [START_REF] Takahashi | Active dendritic currents gate descending cortical outputs in perception[END_REF]. These results highlight the role of pons-projecting ET neurons in specific aspects of motor control.

High-order thalamus

High-order thalamic projections seem to be of an interesting class of ET neurons. Single-cell reconstructions of axons of these ET neurons reveal clusters of axon terminals with large boutons (106, 107). Such giant corticothalamic terminals, so called "driver synapses", elicit large EPSPs and can drive spiking of postsynaptic neurons in the higher-order thalamus (108, 109). Repetitive activation of these synapses results in strong short-term depression, narrowing the temporal window for transmitting activity from the cortex to the thalamus (110, 111). In the somatosensory system, ET neurons in S1 form driver synapses on neurons in the posterior medial nucleus (PoM) of the thalamus. A recent study demonstrated that PoM-targeting ET neurons regulate the mouse's perceptual threshold for detecting tactile stimuli [START_REF] Takahashi | Active dendritic currents gate descending cortical outputs in perception[END_REF]. Importantly, PoM sends its projections back to S1, where they preferentially target IT neurons [START_REF] Petreanu | The subcellular organization of neocortical excitatory connections[END_REF], forming a recurrent loop between the cortex and thalamus (112-114). This recurrent loop may also serve to strengthen the coupling between apical dendrites and soma of L5 neurons [START_REF] Suzuki | General Anesthesia Decouples Cortical Pyramidal Neurons[END_REF].

The motor cortical areas also form a strong recurrent loop with higher-order motor thalamic nuclei.

In a delayed motor task, the loop between the anterior lateral motor cortex (ALM) and the motor thalamus builds persistent preparatory activity reflecting upcoming behavioral choices (115).

Thalamus-targeting ET neurons in ALM therefore play a crucial role in maintaining the premotor activity in the loop.

Striatum

The striatum is a unique structure in the sense that it is innervated by both IT and ET neurons, albeit bilaterally versus unilaterally respectively. Interestingly, in the striatum, axon terminals of IT neurons are characteristically small (0.4-0.5 µm) compared with those of ET neurons (0.8-0.9 µm) (116). This may indicate a difference in the nature of their influence on the striatal activity and an intriguing contrast in their roles during behavior. In the motor system, striatal-targeting IT neurons have been shown to regulate amplitude and speed of goal-directed forelimb movements [START_REF] Park | Motor cortical output for skilled forelimb movement is selectively distributed across projection neuron classes[END_REF]. Moreover, the unidirectional IT-to-ET neuronal connectivity supports the role of IT neurons modulating ET neuronal output and preparing action initiation (103). In the sensory system, during an auditory discrimination task, striatal-targeting IT and ET neurons originating from A1 drive behavioral choices (117). The engagement of striatal-targeting IT and ET neurons is, however, absent during sensory detection or pronounced only to a small extent [START_REF] Takahashi | Active dendritic currents gate descending cortical outputs in perception[END_REF][START_REF] Ruediger | Learning speed and detection sensitivity controlled by distinct cortico-fugal neurons in visual cortex[END_REF], indicating the importance of this pathway for feature discrimination but not for sensory detection. On the contrary to the abovementioned dichotomic roles, there is strong evidence of their participation during sensory associative learning [START_REF] Ruediger | Learning speed and detection sensitivity controlled by distinct cortico-fugal neurons in visual cortex[END_REF]118). Further investigations are necessary to delineate the role of IT-striatal versus ET-striatal pathway in cognitive processes.

CONCLUSION

This review has highlighted differences in morphological, anatomical, and physiological properties between IT and ET neurons, as well as in neuromodulatory effects on cellular excitability. Based on these differences, we hypothesized a distinct role for IT and ET neurons in cortical sensory processing: IT neurons generate an internal model, which acts as a spatiotemporal filter that gates the output of a selected population of ET neurons to the subcortical structures.

With recent methodological advances, such as genetic targeting, population recording and optogenetics, it has now become possible to investigate their modes of operation in the living brain and even during behavior. As a result, distinct modes of operation between subtypes in sensory and motor processing are becoming evident, each impacting behavior in different ways. However, it remains to be determined how the morphological and physiological characteristics contribute to the in vivo operation of each subtype during behavior [START_REF] Takahashi | Active dendritic currents gate descending cortical outputs in perception[END_REF]119). The highlighted difference between IT and ET neurons, as well as the hypothesized computational roles for each subtype, in this review may be useful to facilitate both experimental and theoretical studies to gain a mechanistic understanding of L5 subcircuits in cortical information processing.

In this study, we reviewed IT and ET neurons in neocortical L5, focusing on rodent studies.

Recently, attempts have begun to characterize human L5 neurons physiologically, morphologically and genetically using surgically resected human cortical tissue. These studies have revealed many conserved properties of rodent and human L5 neurons, supporting the translational relevance of the findings in rodents, while some divergent properties have also been identified (120-122). For example, compared to rodents, distal inputs provide very little excitation to the soma in human ET neurons even in the presence of dendritic spikes (122). These examples emphasize the need for continued research identifying conserved and specialized properties of L5 neuronal subtypes in humans and possibly in other animal species. The future studies would provide deeper insights into the fundamental and universal role of L5 neurons in cortical computation, and would also highlight phenotypic divergence related to cortical functions specialized to individual animal species.

Figure 1 .

 1 Figure 1. IT and ET neurons in neocortical L5 | Morphological and electrophysiological

Figure 2 .

 2 Figure 2. Hypothesized circuits and computational roles of IT and ET neurons in L5 | (A) IT

ACKNOWLEDGMENTS

We thank Jiameng Wu, Eduardo Maristany, and Timothy Zolnik for their comments on an earlier version of the manuscript. This study was supported by the University of Bordeaux Initiative of Excellence program (N.T.); the Region Nouvelle-Aquitaine (N.T.); the ATIP-Avenir program (N.T.); the Brain Science Foundation (N.T.); the Einstein Center for Neurosciences Berlin (Ph.D. fellowship and project number A-2021-644) (S.M.); Human Brain Project, EU Commission Grants 945539 (SGA3) (S.M.) and the Deutsche Forschungsgemeinschaft (DFG, SFB1315 and EXC-2049-390688087) (S.M.).

AUTHOR CONTRIBUTIONS

S.M. and N.T. wrote the paper.

DECLARATION OF INTERESTS

The authors declare no competing interests.


Luaï Jaff
email: luai.jaff@gmail.com

Cyrille Bertelle
email: cyrille.bertelle@litislab.eu

Gérard H E Duchamp

Control on System Diffusion Using Genetic Automata

Keywords: Self-organization, Genetic Automata, Diffusion, Control

Complex systems (LeMoigne 1999) are models which describe dynamic organizations or systems where the internal interaction network between their components does not allow the control of individual components with efficience. In a lot of cases, the individual control leads to break the whole structure itself. In other cases, some complex systems exhibit resilience properties which lead them to recover their initial state after small perturbations or individual controls. Controlling complex systems need to not trying to manage the individual controls but to manage the global control, guiding the whole system without directly act on individuals themselves. The study presented in this paper explains how to control a high level system property, the diffusive behavior, by spectral analysis over genetic automata. First experiments are presented using MuPad implementation.

COMPLEX SYSTEMS AND FEED-BACKS FOR ECONOMY

We describe the systemic approach of economical system modelling, especially the feed-back concept which can be considered as a global control over the system components.

Complexity in Economy

We can classify the economical systems in two main classes [START_REF] Marshall | Principles of Economics[END_REF]):

• The classical approaches based on a rational choice of sets of equations representing the state of the economy. The drawbacks from this approach are: the unrealistic assumption of rational choices and perfect information; assumptions on homogeneity and lack of distinction between the agent and the aggregate level; inability to account for the emergence of new kinds of entities, patterns, structures, etc.

• Complexity perspective approach. This approach is more recent as a view of economy. It focusses on viewing the economy as an evolving complex system, using Holland's complex adaptive systems paradigm [START_REF] Holland | Hidden Order -How adaptation builds complexity[END_REF].

With the complexity approach, the aim is to understand how global economic phenomena arise purely from the local interactions and local knowledge of the agents. This stems from the recognition that there is no central or global control in an economy and that the global regularities which arise are purely due to the local interactions of adaptive autonomous agents. The agents within an economic network may be individuals or institutions. We can consider an institution agent as being composed of many individual agents.

Feed-Backs Modelling

W. Brian Arthur [START_REF] Arthur | Complexity and the Economy[END_REF], and other economic theorists have been interested to a view of the economy based on feedbacks, more precisely, positive feedback mechanisms. We can say that modern economies can be divided into two interrelated branches :

• Diminishing Returns and Negative Feedback.

Diminishing returns refers to the notion that the return that a company receives for additional effort decreases as the number of units (outputs) increases.

Diminishing returns explains why industrial companies become more inefficient once they grow over a certain size.

Negative feedback is the process of feeding back to the input a part of a system's output, so as to reverse the direction of change of the output. This tends to keep the output from changing, so it is stabilizing and attempts to maintain constant functionning conditions.

Diminishing returns assumptions are the basis of the conventional economic theory. The actions of the economy induces a negative feedback which leads to predictable equilibrium for prices and market share. Negative feedback stabilizes the economy because any major change will be cancelled out by the reaction they generate. Diminishing returns imply a unique equilibrium point for the economy.

Alfred Marshall [START_REF] Marshall | Principles of Economics[END_REF]) believed that we can not applied the increasing returns everywhere in the economy. His remark leads W. Brian Arthur to observe that the part of the economy that are resource-based (the traditional part) are still subject to diminishing returns, on one hand. And the part of the economy that are knowledge-based (the newer part) are still subject to increasing returns, on the other hand as defined in the following.

• Increasing Returns and Positive Feedback

Increasing returns refer to the notion that the greater the size of the network, the greater the advantage of each participant of the network (network effects). Each participant of the network brings value to the overall network. This is in contrast to diminishing returns which refers to the greater the size (number of users) the less each participant can benefit from participation. Increasing returns generate instability.

We call a feedback mechanism positive if the resulting action goes in the same direction as the condition that triggers it. Positive feedback is an open system contain many types of regulatory systems, among which are systems that involve positive feedback and its relative negative feedback.

Increasing returns make for multiple solutions (multiple equilibrium points), and it generate instability or criticality stability following Per Bak [START_REF] Bak | How nature works -the science of selforganized criticality[END_REF].

After describing the scientific environment of complexity, we give in the next section formal tools allowing to model some auto-regulation processes leading to implement the feed-back mechanisms.

AUTOMATA-BASED SYSTEM MODELLING

We focus our attention on a special kind of automata with outputs which are efficient in operational way [START_REF] Hopcroft | Introduction to automata theory, languages and computation[END_REF]). This automata with output are called automata with multiplicities and they are defined in the following.

Automata with multiplicites

An automaton with multiplicities is based on the fact that the output data of the automata with output belong to a specific algebraic structure, a semi-ring. In that way, we will be able to build effective operations on such automata, using the power of the algebraic structures of the output data. And we are also able to describe this automata in matrix representation with all the power of the linear algebra. Definition 0.1 An automaton with multiplicities over an alphabet and a semi-ring K is the 5-uple Σ, Q, I, T, δ where We remark that

• Q = {S1, S2, ...,
• automata with multiplicities is a generalization of finite automata. In fact, finite automata can be considered as automata with multiplicities with for the semi-ring K, the boolean set B = {0, 1}. To each transition we affect 1 if it exists and 0 if not.

• We have not yet, on purpose, defined what a semiring is. Roughly it is the least structure (K, +, x) that allows matrix computation with units (one can think of a ring without the minus operation)

Linear Representation

The previous automata with multiplicities can be expressed by a linear representation which is a triplet p = (λ, µ, γ)

with λ ∈ K 1×n is a row-vector which coefficients are λ i = I(S i ), γ ∈ K n×1 is a row-vector which coefficients are γ i = T (S i ), and µ : Σ → K n×n is a morphism of monoids such that ∀a ∈ Σ, the coefficient on the i-th row and j-th column of all transitions labeled with a is µ(a) ij = δ(S i , A, S j ).

In the following, we describe the linear representation of the automata with multiplicities which corresponds to the the following figure. In this figure, the input states are characterized with input arrows from nothing and the output states are characterized with output arrow to nothing. On each transition, we indicate before the semi-column, the input data and after the semi-column, the output associated for each input data.

The linear representation is the following λ = 3 0 0

γ t = 0 2 1 µ(a) =   0 0 4 0 0 -1 0 0 -3   µ(b) =   0 3 0 0 0 0 0 0 0   GENETIC AUTOMATA
Genetic algorithms play an important role in the studies of complex adaptive systems, ranging from adaptive agents in economic theory to the use of machine learning techniques. We describe the principles of these algorithms and then their application to genetic automata.

Genetic Algorithms and Operations

The major aspect of genetic algorithms is the adaptation property. This property can be considered as the search of the optimum of a specific function. Genetic algorithms work on a population of individuals. The individuals are represented with chromosomes which are composed with primitive information called alleles.

In term of computable formalization, chromosomes are generally strings or sequences of information over a finite alphabet. To begin the algorithm, we generate a population of chromosomes.

The algorithm is based on a variation-selection process:

• The variation step concerns the basic genetic operators on the individual level and so this step acts on the chromosomes. These basic operators are composed of reproduction/duplication, crossing-over and mutation and we describe how implement them for genetic automata in the following section. The result gives another population with a greater number of chromosomes than the initial one

• The selection step concerns the population level inside which a selection function modifies the whole population constitution. This step leads to keep only some of the chromosomes that have been generated during the variation step and which satisfy to specific constraints

Automata Chromosome Coding and Genetic Operators

We define the chromosome for each automata with multiplicities as the sequence of all the matrices associated to each letter from the (linearly ordered) alphabet. The chromosomes are composed with alleles which are here the lines of the matrices. In the processus, genetic algorithms generate new automata containing possibly new transitions from the

Figure 1 :

 1 Figure 1: Automata with multiplicities

  Figure 2: Chromosome code

  Sn} is the finite set of state• I : Q → K is afunction over the set of initial states, which associates to each initial state a value of K, called entry cost, and to non-initial state a null value

• T : Q → K is a function over the set of the final states, which is associated to each final state a value of K, called final cost, and to non-final state a null value

• δ is the transition function, that is δ : Q × Σ × Q → K which from a

state S1 , a letter a and a state S2 go to a value z of K if it exist a transition labeled with a from the state S1 to the state S2 and with the output z.

ones included in the initial automata. The genetic operators are applied as described in the following:

• Reproduction/duplication: this operator simply copy the automaton chromosome.

• Crossing-over: From 2 chromosomes, it consists in cutting of the initial chromosomes in the same place chosen at random and in permuting the two parts of the matrices lines chains.

• Mutation : In this step, we chose a few chromosomes at random to be candidate for the mutation process. If selected, we modify one line of the matrix lines chromosome. If the matrix lines have to respect some properties (like probabilistic matrix) we constraint the new values to respect them.

EXPERIMENTS ON DIFFUSIVE PROPERTY CONTROL

We present in the following some experiments based on the package MuPAD-Combinat which implements many data structures related to combinatorics [START_REF] Hivert | MuPAD -Combinat, an open-source package for research in algebraic combinatorics[END_REF]. The genetic operators are developped on the predefined domain called Weighted Automaton from this package.

Our experiment consists to define chromosome code for probablitistic automata which are specific automata with multiplicities satisfying the condition that the sum of each lines coefficient must be equal to 1.

Fitness Function and Auto-Regulation Process Based on Spectral Analysis

We define a population of genetic probabilistic automata generated at random and we compute for each matrix, the secondary greatest eigenvalue module (the first one is always equal to 1, because of the probablitistic matrices used, respecting Perron-Frobenius Theorem). The genetic algorithm is described in 1.

Experiment on non Diffusive Systems Control

In the figure 3, we represent a series of histograms computed at different steps where we indicate the number of automata according to their secondary greatest eigenvalue module. With the evolution of the population, we observe that the selection will generate population with chromosome of high secondary greatest eigenvalue module.The x-scale is adaptive to the interval containing all the values of this secondary greatest eignevalue for the automata population.

Algorithm 1: Auto-regulation algorithm for genetic automata population

Generation of P the initial population of probabilistic automata chromosome;

while not end of iterations do for each chromosome c ∈ P do Duplicate c; Update P ;

for each couple of chromosomes (c1, c2) ∈ P do Make crossing-over on (c1, c2); Make mutation on c1 and c2 with low random probability; Update P ;

for each chromosome c ∈ P do Compute the fitness of c as the secondary greatest eigenvalue module; Select from P the half higher fitness population and update P with this selection;

Results analysis and interpretation

As expected by the fitness function, we generate successive populations which are characterized by the convergence process which leads to obtain a population where the greatest secondary eigenvalue module of all automata with multiplicities is equal 1. This processus is an auto-regulation processus on the population itself, managed by a feed-back of the result control on the system itself.

By controlling the generating automata population with eigenvalues module equals 1, we constraint the system to not be dissipative. The spectral analysis and the eigenvalues module give some indications on the dissipative aspect during the evolution. If the eigenvalue modules are less than 1, then we can conclude that a dissipation exists.

The non dissipative evolution process over automata with multiplicities proposed here, is a tool based on the power of the algebraic representation of the automata with multiplicities, using some computation on their matrix representation. So we deal on a dynamic data structure according to the complexity of dissipative process over some self-organized systems. Without managing each matrix coefficient value, we give a kind of control tool according to dissipative process increasing or decreasing.

CONCLUSION AND PERSPECTIVES

In this paper, we present a methodology which allows to control a complex system by a global function which is based on the use of a genetic probabilistic automata population. We scan this population by means of a spectral analysis and control the system by reducing its diffusive properties. The applications to economy can be relevant and we propose by this method, a practical


Hongzhen Wang

Georges Boudebs

Cid B De Araújo

Picosecond cubic and quintic nonlinearity of Lithium Niobate at 532 nm

The nonlinear (NL) optical response of bulk lithium niobate (LiNbO3) was investigated at 532 nm using the second harmonic of a Nd:YAG laser delivering pulses of 12 ps. The experiments were performed using the D4σ method combined with the conventional Zscan technique. Two-and three-photon absorption coefficients equal to 0.27 cm/GW and 2.5 × 10 -26 m 3 /W 2 , respectively, were determined. The NL absorption processes were due to transitions from the valence to the conduction band and to free-carriers absorption. The third-and fifth-order NL refractive indices were 𝑛 2 = (2.5 ± 0.6) × 10 -19 m 2 /W and n 4 < 5.5 × 10 -36 m 4 /W 2 . The present results give support for previous experiments that indicate possible fifth-order processes in bulk samples and channel waveguides fabricated with LiNbO3.

I. INTRODUCTION

Metal-oxides such as Potassium Niobate (KNbO3), Sodium Niobate (NaNbO3), and Lithium Niobate (LiNbO3) crystals present interesting properties such as piezoelectricity, electro-optic and nonlinear (NL) optical behavior. In particular, Lithium Niobate (LiNbO3) has received great attention in the past four decades because of their linear and NL optical properties that are very appropriate for devices based on bulk and surface acoustic wave devices, second harmonic generation, optical waveguide modulators, among other applications [START_REF] Toney | Lithium Niobate Photonics[END_REF][START_REF] Arizmendi | Photonic applications of lithium niobate crystals[END_REF][START_REF]Guided-wave Optoelectronics[END_REF]. Indeed, LiNbO3 based waveguide modulators are the primary candidates for applications in optical fiber telecommunication networks and cable television.

The design and optimization of photonic and electro-optical devices motivated many groups in the past to characterize the optical properties of LiNbO3 from the nearinfrared to the blue range using lasers operating in various temporal regimes. The majority of the experiments were dedicated to the study of the second-and third-order optical susceptibilities, 𝜒 (2) and 𝜒 (3) , respectively. While the works related to the second-order response of LiNbO3 are in reasonable agreement, the measurements related to 𝜒 (3) and high-order nonlinearities still deserve further analysis.

The third-order electronic response of LiNbO3 was first investigated by DeSalvo et al. [START_REF] Desalvo | Infrared to Ultraviolet Measurements of Two-Photon Absorption and n2 in Wide Bandgap Solids[END_REF] using a laser operating at 532 nm (2.34 eV) with pulses of 22 ps, applying the Zscan technique [START_REF] Sheik-Bahae | Sensitive measurement of optical nonlinearities using a single beam[END_REF]. They report a third-order refractive index, 𝑛 2 ∝ 𝑅𝑒𝜒 (3) , of (440±70) × 10 -14 esu, and two-photon absorption (2PA) coefficient, 𝛽 ∝ 𝐼𝑚𝜒 (3) , of (0.38±0.08) cm/GW. One year later, Li et al. [START_REF] Li | Picosecond Z-scan study of bound electronic Kerr effect in LiNbO3 crystal associated with two-photon absorption[END_REF] reported 𝑛 2 = (5.3 ± 0.5) × 10 -19 cm 2 /W and 𝛽 = (0.25 ± 0.6) cm/GW also at 532 nm. Ganeev et al. [START_REF] Ganeev | Characterization of nonlinear optical parameters of KDP, LiNbO3 and BBO crystals[END_REF] working at 532 nm, with pulses of 55 ps, reported 𝑛 2 = 3.4 × 10 -19 m 2 /W for light propagating along the z-axis and 𝛽 = 0.21 cm/GW. More recently, Cherchi et al. [START_REF] Cherchi | Second-harmonic generation in surface periodically poled lithium niobate waveguides: on the role of multiphoton absorption[END_REF] analyzing the efficiency of second harmonic generation in surface periodically-poled LiNbO3 excited by 25 ps pulses in the nearinfrared, observed reduction in the harmonic generation conversion efficiency that was attributed to three-photon absorption (3PA) processes. Working at 770 nm (1.61 eV) the authors estimate a 3PA coefficient, 𝛾 ∝ 𝐼𝑚 𝜒 (5) , of 4.5 × 10 -28 m 3 /W 2 and negligible value of 𝑛 4 ∝ 𝑅𝑒 𝜒 (5) . Three years before, Stepanov et al. [START_REF] Stepanov | Scaling up the energy of THz pulses created by optical rectification[END_REF] investigating THz generation by optical rectification in LiNbO3 excited at 800 nm (1.55 eV), also suggested that 3PA was responsible for the saturation of the THz signal being generated. However Cherchi et al. [START_REF] Cherchi | Second-harmonic generation in surface periodically poled lithium niobate waveguides: on the role of multiphoton absorption[END_REF] and Stepanov et al. [START_REF] Stepanov | Scaling up the energy of THz pulses created by optical rectification[END_REF] could not make precise measurements of the NL absorption coefficients due to the intrinsic limitation of the techniques used.

Beyer et al. [START_REF] Beyer | Femtosecond time-resolved absorption processes in lithium niobate crystals[END_REF] also measured 2PA coefficients of LiNbO3 from 388 to 800 nm, using 0.24 ps laser pulses. In the green region (at 514 nm) it was measured a 2PA coefficient of 0.1 cm/GW. High-order nonlinearities were not investigated in [START_REF] Beyer | Femtosecond time-resolved absorption processes in lithium niobate crystals[END_REF].

The main motivation for the present work was the investigation of the relative contribution of 2PA and 3PA when light at 532 nm is incident on LiNbO3. The experiments were made with laser intensities varying from 10 to 150 GW/cm 2 using the D4σ method [START_REF] Boudebs | Nonlinear characterization of materials using the D4σ method inside a Z-scan 4f-system[END_REF][START_REF] Besse | Picosecond nonlinearity of GeO2-Bi2O3-PbO-TiO2 glasses at 532 and 1,064 nm[END_REF][START_REF] Manzani | Third-order nonlinearties and other properties of molybdenum leadpyrophosphate glass[END_REF][START_REF] Amaral | D4σ curves described analytically through propagation analysis of transverse irradiance moments[END_REF][START_REF] De Araújo | Techniques for nonlinear optical characterization of materials: a review[END_REF] combined with the Z-scan technique that allowed precise NL measurements. This paper is organized as follows: In Section II we present some characteristics of the LiNbO3 sample and a description of the experimental setup as well as the NL technique used. The NL refractive index and NL absorption measurements, at different laser intensities, are presented in Section III with an evaluation of the results. In Section IV a summary of the results and conclusions are presented.

II. EXPERIMENTAL DETAILS

Undoped congruently grown LiNbO3 crystals (z-cut) of good optical quality were obtained from Crystals Technology, Inc. The samples with 0.55 mm thickness have a composition uniformity of (48.38 ± 0.01) mol % of Li2O and present large optical transmission window from ≈315 to ≈4600 nm. For the NL measurements, we used the second harmonic of a Nd: YAG laser at 532 nm (12 ps, 10 Hz). The D4σ -Z-scan technique was applied to characterize the intensity dependent refractive behavior of the samples and the open-aperture Z-scan technique was used to determine the NL absorption coefficients. All measurements were performed at room temperature with linearly polarized light incident perpendicularly to the sample's surface.

The setup used for the D4σ -Z-scan measurements is illustrated in Figure 1. The sample, mounted on a translation stage, is moved along the beam direction (z axis) in the focus region. For each 1 mm step of the motor an image of the light pulses transmitted by the sample is recorded by a CCD sensor. The response of the CCD to optical power was verified to be linear. A second arm was used to monitor the fluctuations of the incident laser pulses (via lens L3). The 4f-imaging system composed of two 20 cm focusing lenses (f1=f2) is described with basis on Fourier optics. Moving the sample in the focal region allows determination of the material's NL parameters because the induced NL phase-shift and/or NL absorption originate changes in the transmitted beam profile. The imaging system is aligned to obtain a magnification equal to 1.0 such that it is possible to characterize accurately the spatial profile of the beam at the entry of the experimental setup, in the linear regime (for low incident laser intensity). By using the optical transfer function for the free-space propagation over a finite distance and considering the phase transformations due to the 4f-system, we simulate the propagation of the beam from the object to the image plane taking into account the transmittance of the NL medium positioned at each motor step used for the Z-scan measurement as in [START_REF] Boudebs | Investigations on the nonlinear optical response and losses of toluene at 532 nm and 1064 nm in the picosecond regime[END_REF]. The samples' transmittance is considered according to the evolution equations for the NL phase-shift and intensity in the thin sample approximation. Fits of the beam waist relative variation (BWRV) spatial profile of the output beam provide the NL refractive index of the sample by comparing the numerical and experimental data. On the other hand, the NL absorption coefficient is obtained from the fits of the conventional open aperture Z-scan profile. For each measurement, two sets of acquisitions are performed. The first set is in the NL regime and the second one in the linear regime, obtained by reducing the incident laser intensity; this is necessary to remove from the NL measurements the diffraction, diffusion and/or imperfection contributions due to sample inhomogeneity. Moreover, as previously mentioned, an image of the entry plane of the 4f-system allowed the characterization of the circular aperture object, with approximately 1.6 mm radius, that is considered in the computer simulation. As shown in [START_REF] Boudebs | Investigations on the nonlinear optical response and losses of toluene at 532 nm and 1064 nm in the picosecond regime[END_REF] this procedure allows high accuracy measurements even for relatively high absorption and large phase-shift when the BWRV signal is no more linear with the incident laser intensity. The main source of uncertainty in the present experiments is due to the absolute measurement of the laser pulse energy because the accuracy of the joulemeter used is about 10%. To be selfconsistent, in the D4σ method we use a top-hat beam in the 4f-imaging system and we acquired, on the CCD, the intensity profile I(x,y) after the interaction of the light beam with the NL material. The acquisitions obtained for each z position allow to compute the first-and the second-order moments of I(x,y), therefore, the broadening and the narrowing of the aperture radius versus z. The method gives four times the standard deviation of the spatially limited intensity distribution. The reading of the image through a CCD allows correction for point-beam instability following the centroid of the image position while calculating the first-order moment of the intensity distribution.

The simulation of the whole image formation processing gives the NL parameters even in the presence of high NL absorption. No simplified formulas are used. More details on the D4σ method are given in refs. [START_REF] Boudebs | Nonlinear characterization of materials using the D4σ method inside a Z-scan 4f-system[END_REF][START_REF] Besse | Picosecond nonlinearity of GeO2-Bi2O3-PbO-TiO2 glasses at 532 and 1,064 nm[END_REF][START_REF] Manzani | Third-order nonlinearties and other properties of molybdenum leadpyrophosphate glass[END_REF][START_REF] Amaral | D4σ curves described analytically through propagation analysis of transverse irradiance moments[END_REF][START_REF] De Araújo | Techniques for nonlinear optical characterization of materials: a review[END_REF][START_REF] Boudebs | Investigations on the nonlinear optical response and losses of toluene at 532 nm and 1064 nm in the picosecond regime[END_REF].

We remark that an advantage of using top hat beams (binary circular objects) is to be sure about the beams spatial extension while defining the object in the data acquisition program. Gaussian beams are less accurate to determine because of the inherent noise that comes with the images given by the CCD in which the wings of such beams are embedded into noise. Using circular binary objects allows to remove all the noisy pixels where the intensity is less than a treshold value of the normalized image profile because no light is transmitted outside the circular aperture. Note that if the threshold is set to a given value, the normalized object should not contain areas where the intensity is lower than this value. It is important to remark that the simplified formulas used for some kind of laser beams (such as for Gaussian beams in [START_REF] Boudebs | Nonlinear characterization of materials using the D4σ method inside a Z-scan 4f-system[END_REF]) can not be used here due to the relatively high NL response observed in the present experiments.

III. RESULTS AND DISCUSSION

A. NL coefficients measurements considering the third-order response

The classical measurement procedure was built to consider the response of the material described by an effective cubic nonlinearity defined by 2 C (m/W), the third- order NL absorption coefficient, and 2eff n (m 2 /W), the effective NL refractive index. In these conditions, the amplitude transmittance of the sample is described by

( ) ( ) ( ) ( ) eff I z, u, v I z, u, v log 1 q z, u, v q z, u, v =+  
. The peak on-axis NL phase-shift at the focus is NL eff 0 (0, 0, 0)  =  which can be deduced from the BWRV signals (for more details see [START_REF] Boudebs | Nonlinear characterization of materials using the D4σ method inside a Z-scan 4f-system[END_REF]). To obtain n 2eff , it is necessary to take into account the actual NL absorption measured at this intensity level considering C2.

The results obtained by the fittings are represented by the solid lines (blue color) in Figure 3). The results are summarized in Figure 4. The intensity dependent value of C2 and the imperfect agreement of the 2PA fittings shown in Figure 3 motivated us to investigate higher order NL absorption as will be presented in the next subsection.

B. NL absorption measurements considering higher-order response

We assume that, under the slowly varying envelope and thin sample approximations, the evolution of the optical intensity, I, as a function of 𝑧 ′ , the light propagation axis inside the material, obeys the equation:

dI dz ′ = -C N I N (1)
where N is an integer to be determined. Equation ( 1) is identical to the evolution equation for the intensity when the losses are due to N-photon absorption (NPA) with CN being the corresponding NL absorption coefficient. Considering the boundary conditions I(z ′ = 0) = I 0 (z, u, v, t) and I(z ′ = L) = I L (z, u, v, t), the solution of Equation ( 1) is:

I L (z, u, v, t) = I 0 (z,u,v,t) {1+(N-1)C N L[I 0 (z,u,v,t)] N-1 } 1 (N-1) ⁄ (2)
Of course for N=2, Eq (2) reduces to the squared modulus of the amplitude transmittance T defined in the subsection A.

For the open-aperture Z-scan profile, Equation ( 2) is integrated numerically over space and time (over u, v and t) assuming a Gaussian temporal profile and the actual spatial profile of the beam at the entry (after propagation to the z position in the focal region). The numerical integration at this level for each z position using Equation ( 2) is advantageous because it avoids approximations that assume relatively low NL absorption correction in the far-field, i.

C. Co -existence of 2PA and 3PA processes

LiNbO3 exhibits a wide transparency window in the visible but there is a strong absorption band from ≈4.2 to ≈7.3 eV [START_REF] Mamoun | Energy band gap and optical properties of lithium niobate from ab initio calculations[END_REF][START_REF] Riefer | Lithium Niobate Dielectric Function and Second-Order Polarizability Tensor from Massively Parallel Ab Initio Calculations[END_REF]. Therefore, two-and three-photon absorption processes are expected by considering the large intensities of the incident laser with wavelength 532 nm (2.34 eV). For this reason the data in Figure 3 at lowintensity was well described by 2PA (the blue solid line) considering C2 = 0.27 cm/ GW, while at moderate and large intensities the data is better adjusted by considering 3PA (the red dashed line) as the dominant process with C3=2.5 × 10 -26 m 3 /W 2 .

Figure 5 shows the intensity dependence of C3 which results in a constant value particularly for intensities larger than 5 × 10 14 W/m 2 . The relative contribution of the 2PA and 3PA processes can be evaluated in the following way. First, we recall that the best-fit line is the line that minimizes the squared deviations from theory given by E = ∑ (T ei -T pi ) 2 i

, where T pi is the predicted value on the line for a given number position i, and Tei is the actual value measured for that given i. Then, we defined the Mean Vertical Deviation as MVD = E N with N being the total number of acquisitions (generally 61) for a given intensity corresponding to the Z-scan profile. The intensity dependence of MVD is illustrated by Figure 6. Notice that the MVD globally increases with the laser intensity when considering the 2PA process while the fitting is much better considering 3PA as the main contribution for high and moderate intensities (see the dashed lines in Figure 3). Therefore, the results shown in the Figure 3 and Figure 4 indicate that the two NL absorption processes, 2PA and 3PA, occur for intensities lower than 5 x 10 14 W/m 2 but the 2PA process is dominant. The 3PA process is dominant for laser intensities larger than 5 x 10 14 W/m 2 because the electron population in the conduction band becomes large and therefore the one-photon free-carriers absorption becomes relevant. That may explain the different conclusions by the authors of [START_REF] Li | Picosecond Z-scan study of bound electronic Kerr effect in LiNbO3 crystal associated with two-photon absorption[END_REF][START_REF] Ganeev | Characterization of nonlinear optical parameters of KDP, LiNbO3 and BBO crystals[END_REF] and [START_REF] Cherchi | Second-harmonic generation in surface periodically poled lithium niobate waveguides: on the role of multiphoton absorption[END_REF][START_REF] Stepanov | Scaling up the energy of THz pulses created by optical rectification[END_REF].

IV. SUMMARY

In the present paper we demonstrate that congruent LiNbO3 in the presence of strong picosecond lasers operating at 532 nm may present two-photon absorption from the valence to the first conduction band and three-photon absorption. The three-photon absorption process is a two-step excitation process (two-photon absorption without real intermediate state to the first conduction band followed by one-photon absorption by free-carriers). The 2PA coefficient C2 = 0.27 cm/GW was determined at relatively lowintensity while at moderate and large intensities the dominant process was found to be 3PA with C3=2.5 × 10 -26 m 3 /W 2 . The third-and fifth-order NL refractive index were respectively 𝑛 2 = (2.5 ± 0.6) × 10 -19 m 2 /W and n 4 < 5.5 × 10 -36 m 4 /W 2 .

Fig. 1 .

 1 Fig.1. The 4f-imaging system. The sample is moved along the beam direction around

  z represents the position of the sample in the focus region. u=x/λf1 and v=y/λf1 are the normalized spatial frequencies with  as the incident laser wavelength. u, v C LI z, u, v = where L denotes the thickness of the sample and I(z, u, v) is the intensity of the laser beam

Figure 2

 2 Figure 2 for the NL refraction and Figure 3 for the NL absorption for three different

Fig. 2 .Fig. 3 (

 23 Fig. 2. (Color online) The NL refraction signal associated to the beam waist relative

Fig. 4 (

 4 Fig. 4 (Color online) Evolution of the measured effective third-order coefficients as a

Figure 4 (

 4 Figure 4(a) shows the behavior of n 2eff for the various intensities showing a constant value whose average is (2.5 ± 0.6) × 10 -19 m 2 /W. Taking into account the

Figure 4 (

 4 Figure 4(b) summarizes the behavior of C 2 showing a noticeable increase versus the incident intensity. The average value of C 2 calculated up to intensities of 1.5 × 10 15 W/m 2 is (3.5 ± 0.7) × 10 -12 m/W in good agreement with [6, 7]. A comparison

  e.: |(N -1)C N L[I 0 (z, u, v, t)] N-1 | ≤ 1. Thus, performing the open-aperture Z-scan experiment, the value of CN can be deduced, for any regime by fitting the normalized transmittance data profile.

Fig. 5 .

 5 Fig. 5. (Color online) Measured 3PA coefficient, C3, as a function of the incident

Fig. 6 (

 6 Fig. 6 (Color online) The mean value of the vertical deviations between the data

ACKNOWLEDGEMENTS

This work was performed in the framework of the National Institute of Photonics (INCT de Fotônica) project. We acknowledge financial support from the Brazilian agencies Conselho Nacional de Desenvolvimento Cientifico e Tecnológico (CNPq) and the Fundação de Amparo à Ciência e Tecnologia do Estado de Pernambuco (FACEPE). Cid B. de Araújo appreciates the hospitality of the Université d'Angers where the experiments were made and the financial support from the LUMOMAT and the NNN-TELECOM programs.