Localization of neurons has the promise boosting neuronal trace and quantifying the large scale neuronal circuits. In this paper, we have proposed NeuroGPS method to locate neurons across different brain areas and have demonstrated its high robustness to the shape, size and spatial distribution of neurons. Specifically, this method eliminated the negative influence of the complicated neurites, especially from the thick dendritic truck, on localization.
Essentially, we made a new biophysical model to locate the neuron which only concerns the morphology of the neuronal soma, rather than the entire neuron. In the biophysical model, we introduced L1 minimization to maximize image sparsity, and identify the false positive positions in thick dendritic truck which is based on the biophysical/neurobiological assumption that each neuron has only one soma that does not overlap with its neighbors. This biophysical model and its successful solution play a key role in successfully locating neurons with complicated morphology.
Due to the complexity and diversity of neuronal structure in mammals like mouse, neurons across different brain have a wide variation of shape, size and spatial distribution. Therefore localization of these neurons without human interference is a challenging work. Typically, when we analyzed our data sets, the radiuses of some thick trucks are close to 4.0 μm (), which are almost equals to or even bigger than the radius of some neuronal somas ( & ). Previous works17,18,19,20,21,22,23,24,25,26,27,28,29
have not involved this problem, and thus experience difficulties in resolving it. For example, though the classical method like FARSIGHT can overcome the challenges in locating neurons with variation of size and spatial distribution25
, it cannot distinguish thick trucks and some small neurons (). NeuroGPS, as described earlier, can induce shrinkage in the radiuses of thick trucks by introducing L1-M and effectively identify the false positions of thick trucks ( & ). In addition, NeuroGPS possesses high robustness to the size and spatial distribution of neurons ( & ). These advantages of NeuroGPS make it possible to locate neurons across different brain areas.
In optimization problem (4), the parameter λ is used to control the tradeoff between the fitting error and the sparsity of radiuses of sphere function. When we choose a big value of λ to enforce this sparsity, the fitting error will increase. Conversely, the small fitting error is built on the sacrifice of the sparsity. So finding the appropriate tradeoff may be important for the localization performance. Fortunately, the algorithm of reweighted L1 minimization34
can automatically modify the strength of the sparsity in solving optimization problem (4) and provide strong sparsity even when a small value of λ is set, e.g. strong sparsity can be gained with a wide λ value range. This feature enables that the localization performance is robust to the tradeoff λ. In experiment, specific λ value was roughly estimated by analyzing a few images. If the localization result is right, the corresponding λ is set, and then extend to the rest volume datasets. If the result is not right, tune λ again to get better performances. In our experience, λ remains unchanged for any image stacks provided that the size of volume pixels and the radius of the soma are within a certain range.
In our experimental data sets analysis, a small amount of positions of neurons has not been detected as expected. There are some reasons for this phenomenon. Firstly, in the process of sample preparing or imaging, the shape of some neuronal soma may be seriously distorted and deviated far from its normal morphology. Secondly, information loss from neuronal somas in the binarization and erosion operation has a negative influence on seeds selection and sphere functions fitting. Depending on the feature of neuronal morphology, using the improved binarization and erosion method may improve the localization performance.
The computation time of NeuroGPS depends on the number, morphology and spatial distribution of neurons. As described earlier, we used the averaging method combined with gradient projection to speed the iteration optimization process and increase the computation efficiency of NeuroGPS. This operation is effective based on the fact the shape of neuronal soma basically meets spherical symmetry. As verified by Berglund36
, an object can be located by averaging method provided that its shape meets the spherical symmetry. Typically, analyzing an 600 × 200 × 75 volume pixels image stacks takes about 30 seconds in an Inter(R)Xeon(R)CPU 3.46 GHz computing platform. On the other hand, NeuroGPS performs independent optimizations on different extracted regions, so it can be easily implemented on massive parallel computing for more enhanced speed.
Although we built this NeuroGPS using fluorescent brain image stacks, it can also be applicable to other kinds of images, for example, Nissl staining neuronal images (576 × 1623 × 20 volume pixels). NeuroGPS well located these neurons with the false positive rate of 6% and true positive rate of 93% compared with manual detected positions. Interestingly, the signal intensity of neuronal soma in this example is not uniform. In this case, NeuroGPS is still effective. These results may indicate that NeuroGPS can be applied in a variety of data sets.
It should be noted that our NeuroGPS only focus on the localization of neurons, and cannot be implemented on segmentation of neurons at present. The main reason is that the complicated morphology and the dense distribution of neurons make high-accuracy segmentation of neurons become an acknowledged difficulty4,37,38
, beyond our current concerns. Nevertheless, NeuroGPS provide a tool to automatically and accurately locate neurons, which is helpful for neuronal dendritic tracing. Some newly developed neuron tracing methods7,8,9
experience difficulties in rejecting the neuronal soma interference. Future work is expected to combine NeuroGPS with automatic segmentation method and facilitating automatic neuronal circuits tracing.