Papers by leonardo franco
The firing of inferior temporal cortex neurons is tuned to objects and faces, and in a complex sc... more The firing of inferior temporal cortex neurons is tuned to objects and faces, and in a complex scene, their receptive fields are reduced to become similar to the size of an object being fixated. These two properties may underlie how objects in scenes are encoded. An alternative hypothesis suggests that visual perception requires the binding of features of the visual target through spike synchrony in a neuronal assembly. To examine possible contributions of firing synchrony of inferior temporal neurons, we made simultaneous recordings of the activity of several neurons while macaques performed a visual discrimination task. The stimuli were presented in either plain or complex backgrounds. The encoding of information of neurons was analyzed using a decoding algorithm. Ninety-four percent to 99% of the total information was available in the firing rate spike counts, and the contribution of spike timing calculated as stimulus-dependent synchronization (SDS) added only 1-6% of information to the total that was independent of the spike counts in the complex background. Similar results were obtained in the plain background. The quantitatively small contribution of spike timing to the overall information available in spike patterns suggests that information encoding about which stimulus was shown by inferior temporal neurons is achieved mainly by rate coding. Furthermore, it was shown that there was little redundancy (6%) between the information provided by the spike counts of the simultaneously recorded neurons, making spike counts an efficient population code with a high encoding capacity.
A local unsupervised processing stage is inserted within a neural network constructed to recogniz... more A local unsupervised processing stage is inserted within a neural network constructed to recognize facial expressions. The stage is applied in order to reduce the dimensionality of the input data while preserving some topological structure. The receptive fields of the neurons in the first hidden layer self-organize according to a local energy function, taking into account the variance of the input pixels. There is just one synapse going out from every input pixel and these weights, connecting the first two layers, are trained with a hebbian algorithm. The structure of the network is completed with specialized modules, trained with backpropagation, that classify the data into the different expression categories. Thus, the neural net architecture includes 4 layers of neurons, that we train and test with images from the Yale Faces Database. We obtain a generalization rate of ± on unseen faces, similar to the ¿ ¾± rate obtained when using a similar system but implementing PCA processing at the initial stage. ¼ ½¼. Since the backpropagation training is an on-line procedure, at the end of the training phase the average error per example is decreased to ¼ ¼¾, approximately.
Neurocomputing, 2006
We introduce a measure for the complexity of Boolean functions that is highly correlated with the... more We introduce a measure for the complexity of Boolean functions that is highly correlated with the generalization ability that could be obtained when the functions are implemented in feedforward neural networks. The measure, based on the calculation of the number of neighbour examples that differ in their output value, can be simply computed from the definition of the functions, independently of their implementation. Numerical simulations performed on different architectures show a good agreement between the estimated complexity and the generalization ability and training times obtained. The proposed measure could help as a useful tool for carrying a systematic study of the computational capabilities of network architectures by classifying in an easy and reliable way the Boolean functions. Also, based on the fact that the average generalization ability computed over the whole set of Boolean functions is 0.5, a very complex set of functions was found for which the generalization ability is lower than for random functions. r
Physica A-statistical Mechanics and Its Applications, 2004
We analize the statistical mechanics of a long-range antiferromagnetic model defined on a Ddimens... more We analize the statistical mechanics of a long-range antiferromagnetic model defined on a Ddimensional hypercube, both at zero and finite temperatures. The associated Hamiltonian is derived from a recently proposed complexity measure of Boolean functions, in the context of neural networks learning processes. We show that, depending of the value of D, the system either presents a low temperature antiferromagnetic stable phase or the global antiferromagnetic order disappears at any temperature. In the last case the ground state is an infinitely degenerated non-glassy one, composed by two equal size anti-aligned antiferromagnetic domains. We also present some results for the ferromagnetic version of the model.
IEEE Transactions on Neural Networks, 2001
The parity function is one of the most used Boolean function for testing learning algorithms beca... more The parity function is one of the most used Boolean function for testing learning algorithms because both of its simple definition and its great complexity. Being one of the hardest problems, many different architectures have been constructed to compute parity, essentially by adding neurons in the hidden layer in order to reduce the number of local minima where gradient-descent learning algorithms could get stuck. We construct a family of modular architectures that implement the parity function in which, every member of the family can be characterized by the fan-in max of the network, i.e., the maximum number of connections that a neuron can receive. We analyze the generalization ability of the modular networks first by computing analytically the minimum number of examples needed for perfect generalization and second by numerical simulations. Both results show that the generalization ability of these networks is systematically improved by the degree of modularity of the network. We also analyze the influence of the selection of examples in the emergence of generalization ability, by comparing the learning curves obtained through a random selection of examples to those obtained through examples selected accordingly to a general algorithm we recently proposed.
Neurocomputing, 1998
We design new feed-forward multi-layered neural networks which perform di erent elementary arithm... more We design new feed-forward multi-layered neural networks which perform di erent elementary arithmetic operations, such as bit shifting, addition of N p -bit numbers, and multiplication of two n-bit numbers. All the structures are optimal in depth and are polinomialy bounded in the number of neurons and in the number of synapses. The whole set of synaptic couplings and thresholds are obtained exactly.
We analize the statistical mechanics of a long-range antiferromagnetic model defined on a D-dimen... more We analize the statistical mechanics of a long-range antiferromagnetic model defined on a D-dimensional hypercube, both at zero and finite temperatures. The associated Hamiltonian is derived from a recently proposed complexity measure of Boolean functions, in the context of neural networks learning processes. We show that, depending of the value of D, the system either presents a low temperature antiferromagnetic stable phase or the global antiferromagnetic order disappears at any temperature. In the last case the ground state is an infinitely degenerated non-glassy one, composed by two equal size anti-aligned antiferromagnetic domains. We also present some results for the ferromagnetic version of the model.
Neural Computation, 2000
In this work, we study how the selection of examples a ects the learning procedure in a boolean n... more In this work, we study how the selection of examples a ects the learning procedure in a boolean neural network and its relationship with the complexity of the function under study and its architecture.
Neural Processing Letters, 2009
This work analyzes the problem of selecting an adequate neural network architecture for a given f... more This work analyzes the problem of selecting an adequate neural network architecture for a given function, comparing existing approaches and introducing a new one based on the use of the complexity of the function under analysis. Numerical simulations using a large set of Boolean functions are carried out and a comparative analysis of the results is done according to the architectures that the different techniques suggest and based on the generalization ability obtained in each case. The results show that a procedure that utilizes the complexity of the function can help to achieve almost optimal results despite the fact that some variability exists for the generalization ability of similar complexity classes of functions.
IEEE Transactions on Circuits and Systems I-regular Papers, 2008
A new algorithm for obtaining efficient architectures composed of threshold gates that implement ... more A new algorithm for obtaining efficient architectures composed of threshold gates that implement arbitrary Boolean functions is introduced. The method reduces the complexity of a given target function by splitting the function according to the variable with the highest influence. The procedure is iteratively applied until a set of threshold functions is obtained, leading to reduced depth architectures, in which
Journal of Neurophysiology, 2009
Rolls ET, Grabenhorst F, Franco L. Prediction of subjective affective state from brain activation... more Rolls ET, Grabenhorst F, Franco L. Prediction of subjective affective state from brain activations15 . ing and information theoretic techniques were used to analyze the predictions that can be made from functional magnetic resonance neuroimaging data on individual trials. The subjective pleasantness produced by warm and cold applied to the hand could be predicted on single trials with typically in the range 60 -80% correct from the activations of groups of voxels in the orbitofrontal and medial prefrontal cortex and pregenual cingulate cortex, and the information available was typically in the range 0.1-0.2 (with a maximum of 0.6) bits. The prediction was typically a little better with multiple voxels than with one voxel, and the information increased sublinearly with the number of voxels up to typically seven voxels. Thus the information from different voxels was not independent, and there was considerable redundancy across voxels. This redundancy was present even when the voxels were from different brain areas. The pairwise stimulus-dependent correlations between voxels, reflecting higher-order interactions, did not encode significant information. For comparison, the activity of a single neuron in the orbitofrontal cortex can predict with 90% correct and encode 0.5 bits of information about whether an affectively positive or negative visual stimulus has been shown, and the information encoded by small numbers of neurons is typically independent. In contrast, the activation of a 3 ϫ 3 ϫ 3-mm voxel reflects the activity of ϳ0.8 million neurons or their synaptic inputs and is not part of the information encoding used by the brain, thus providing a relatively poor readout of information compared with that available from small populations of neurons.
Experimental Brain Research, 2004
A new decoding method is described that enables the information that is encoded by simultaneously... more A new decoding method is described that enables the information that is encoded by simultaneously recorded neurons to be measured. The algorithm measures the information that is contained not only in the number of spikes from each neuron, but also in the cross-correlations between the neuronal firing including stimulus-dependent synchronization effects. The approach enables the effects of interactions between the ‘signal’ and ‘noise’ correlations to be identified and measured, as well as those from stimulus-dependent cross-correlations. The approach provides an estimate of the statistical significance of the stimulus-dependent synchronization information, as well as enabling its magnitude to be compared with the magnitude of the spike-count related information, and also whether these two contributions are additive or redundant. The algorithm operates even with limited numbers of trials. The algorithm is validated by simulation. It was then used to analyze neuronal data from the primate inferior temporal visual cortex. The main conclusions from experiments with two to four simultaneously recorded neurons were that almost all of the information was available in the spike counts of the neurons; that this Rate information included on average very little redundancy arising from stimulus-independent correlation effects; and that stimulus-dependent cross-correlation effects (i.e. stimulus-dependent synchronization) contribute very little to the encoding of information in the inferior temporal visual cortex about which object or face has been presented.
Journal of Neurophysiology, 2005
Rolls, Edmund T., Jianzhong Xiang, and Leonardo Franco. Object, space, and object-space represent... more Rolls, Edmund T., Jianzhong Xiang, and Leonardo Franco. Object, space, and object-space representations in the primate hippocampus. . A fundamental question about the function of the primate including human hippocampus is whether object as well as allocentric spatial information is represented. Recordings were made from single hippocampal formation neurons while macaques performed an object-place memory task that required the monkeys to learn associations between objects and where they were shown in a room. Some neurons (10%) responded differently to different objects independently of location; other neurons (13%) responded to the spatial view independently of which object was present at the location; and some neurons (12%) responded to a combination of a particular object and the place where it was shown in the room. These results show that there are separate as well as combined representations of objects and their locations in space in the primate hippocampus. This is a property required in an episodic memory system, for which associations between objects and the places where they are seen are prototypical. The results thus provide an important advance by showing that a requirement for a human episodic memory system, separate and combined neuronal representations of objects and where they are seen "out there" in the environment, is present in the primate hippocampus.
This chapter presents and discusses several well-known constructive neural network algorithms sui... more This chapter presents and discusses several well-known constructive neural network algorithms suitable for constructing feedforward architectures aiming at classification tasks involving two classes. The algorithms are divided into two different groups: the ones directed by the minimization of classification errors and those based on a sequential model. In spite of the focus being on two-class classification algorithms, the chapter also briefly comments on the multiclass versions of several two-class algorithms, highlights some of the most popular constructive algorithms for regression problems and refers to several other alternative algorithms.
The generalization ability of different sizes architectures with one and two hidden layers traine... more The generalization ability of different sizes architectures with one and two hidden layers trained with backpropagation combined with early stopping have been analyzed. The dependence of the generalization process on the complexity of the function being implemented is studied using a recently introduced measure for the complexity of Boolean functions. For a whole set of Boolean symmetric functions it is found that large neural networks have a better generalization ability on a large complexity range of the functions in comparison to smaller ones and also that the introduction of a small second hidden layer of neurons further improves the generalization ability for very complex functions. Quasi-random generated Boolean functions were also analyzed and we found that in this case the generalization ability shows small variability across different network sizes both with one and two hidden layer network architectures.
Constructive neural network algorithms suffer severely from overfitting noisy datasets as, in gen... more Constructive neural network algorithms suffer severely from overfitting noisy datasets as, in general, they learn the set of examples until zero error is achieved. We introduce in this work a method for detect and filter noisy examples using a recently proposed constructive neural network algorithm. The method works by exploiting the fact that noisy examples are harder to be learnt, needing a larger number of synaptic weight modifications than normal examples. Different tests are carried out, both with controlled experiments and real benchmark datasets, showing the effectiveness of the approach.
Breast Cancer Research and Treatment, 2005
The objective of this study is to compare the predictive accuracy of a neural network (NN) model ... more The objective of this study is to compare the predictive accuracy of a neural network (NN) model versus the standard Cox proportional hazard model. Data about the 3811 patients included in this study were collected within the ‘El Álamo’ Project, the largest dataset on breast cancer (BC) in Spain. The best prognostic model generated by the NN contains as covariates age, tumour size, lymph node status, tumour grade and type of treatment. These same variables were considered as having prognostic significance within the Cox model analysis. Nevertheless, the predictions made by the NN were statistically significant more accurate than those from the Cox model (p<0.0001). Seven different time intervals were also analyzed to find that the NN predictions were much more accurate than those from the Cox model in particular in the early intervals between 1–10 and 11–20 months, and in the later one considered from 61 months to maximum follow-up time (MFT). Interestingly, these intervals contain regions of high relapse risk that have been observed in different studies and that are also present in the analyzed dataset.
Biological Cybernetics, 2004
The encoding of information by populations of neurons in the macaque inferior temporal cortex was... more The encoding of information by populations of neurons in the macaque inferior temporal cortex was analyzed using quantitative information-theoretic approaches. It was shown that almost all the information about which of 20 stimuli had been shown in a visual fixation task was present in the number of spikes emitted by each neuron, with stimulus-dependent cross-correlation effects adding for most sets of simultaneously recorded neurons almost no additional information. It was also found that the redundancy between the simultaneously recorded neurons was low, approximately 4% to 10%. Consistent with this, a decoding procedure applied to a population of neurons showed that the information increases approximately linearly with the number of cells in the population.
Biological Cybernetics, 2007
The sparseness of the encoding of stimuli by single neurons and by populations of neurons is fund... more The sparseness of the encoding of stimuli by single neurons and by populations of neurons is fundamental to understanding the efficiency and capacity of representations in the brain, and was addressed as follows. The selectivity and sparseness of firing to visual stimuli of single neurons in the primate inferior temporal visual cortex were measured to a set of 20 visual stimuli including objects and faces in macaques performing a visual fixation task. Neurons were analysed with significantly different responses to the stimuli. The firing rate distribution of 36% of the neurons was exponential. Twenty-nine percent of the neurons had too few low rates to be fitted by an exponential distribution, and were fitted by a gamma distribution. Interestingly, the raw firing rate distribution taken across all neurons fitted an exponential distribution very closely. The sparseness a s or selectivity of the representation of the set of 20 stimuli provided by each of these neurons (which takes a maximal value of 1.0) had an average across all neurons of 0.77, indicating a rather distributed representation. The sparseness of the representation of a given stimulus by the whole population of neurons, the population sparseness a p, also had an average value of 0.77. The similarity of the average single neuron selectivity a s and population sparseness for any one stimulus taken at any one time a p shows that the representation is weakly ergodic. For this to occur, the different neurons must have uncorrelated tuning profiles to the set of stimuli.
Journal of Vascular Surgery, 2006
Objective: To identify the best method for the prediction of postoperative mortality in individua... more Objective: To identify the best method for the prediction of postoperative mortality in individual abdominal aortic aneurysm surgery (AAA) patients by comparing statistical modelling with artificial neural networks' (ANN) and clinicians' estimates. Methods: An observational multicenter study was conducted of prospectively collected postoperative Acute Physiology and Chronic Health Evaluation II data for a 9-year period from 24 intensive care units (ICU) in the Thames region of the United Kingdom. The study cohort consisted of 1205 elective and 546 emergency AAA patients. Four independent physiologic variables-age, acute physiology score, emergency operation, and chronic health evaluation-were used to develop multiple regression and ANN models to predict in-hospital mortality. The models were developed on 75% of the patient population and their validity tested on the remaining 25%. The results from these two models were compared with the observed outcome and clinicians' estimates by using measures of calibration, discrimination, and subgroup analysis. Results: Observed in-hospital mortality for elective surgery was 9.3% (95% confidence interval [CI], 7.7% to 11.1%) and for emergency surgery, 46.7% (95% CI, 42.5 to 51.0%). The ANN and the statistical models were both more accurate than the clinicians' predictions. Only the statistical model was internally valid, however, when applied to the validation set of observations, as evidenced by calibration 14.97; P ؍ .060), discrimination properties (area under receiver operating characteristic curve, 0.869; 95% CI, 0.824 to 0.913), and subgroup analysis. Conclusions: The prediction of in-hospital mortality in AAA patients by multiple regression is more accurate than clinicians' estimates or ANN modelling. Clinicians can use this statistical model as an objective adjunct to generate informed prognosis. ( J Vasc Surg 2006;43:467-73.)
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Papers by leonardo franco