Systems neuroscience studies systems, such as the visual sensory system - from the light (input) to the cortical representation. Systems neuroscience generally implies an integrated approach that includes fundamental or basic neuroscience - the study of cellular and molecular functions of the nervous system(s) - and typically borders to behavioral or cognitive neuroscience.
Whereas cognitive neuroscience aims to explain the "biology of the mind" - for example, how is memory realized in the brain - systems neuroscience aims to explain how the brain works - for example, how does the brain give rise to vision.
Computational neuroscience is fundamentally a mathematical discipline that is realized using computers. It is used to both simulate (theoretical neuroscience) and validate in vivo and in vitro findings in silico. It includes the abstraction of general laws based (more or less) on empirical data. Generally, initial parameters are based on findings from basic neuroscience, which enables biophysically realistic models based on the functionality of neurons to be implemented. However, less realistic simplified models are often used to model neural networks that have an applied AI task. However, these models border to computer science and in particular AI and machine learning.
Neuroscience is a large field and there is a strong need to bridge the different disciplines that contribute to the science, not only through computational neuroscience, but also through neuroinformatics:
The exponential growth of the data produced in neuroscience mean that
computational approaches are fundamental to make sense of this information, including theoretical modelling and methods for large scale data analysis/integration that are capable of handling and comparing data generated over such
diverse timescales (e.g. synaptic versus genetic changes). As such, quantitative theoretical neuroscience must be gradually integrated with
mainstream experimental neuroscience.
—European Brain Council Consensus Statement - March 2016
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"Basic neuroscience" requires an invasive study of neurons in order to study the molecular mechanisms of the cell - on the most basic level. That is why mostly animals are used in this type of study. (However, neuroanatomists also study human post-mortem brains, and molecular imaging can be used in humans, but on a different level of explanation - a whole-brain perspective - with less specificity of the causal mechanisms on the cellular level.) Molecular neuroscience studies, for example, the properties of different types of calcium channels. A famous computational model is the Hodgkin-Huxley model that is based on the study of aplysia. In this regard, basic neuroscience informs computational models that are bottom-up, but there are also top-down models that rely on pure psychology, and a range of models in-between.
N.B. I would be careful to use the term "exploratory", because this implies that the science proceeds without a hypothesis. That is rarely the case in the natural sciences, but some of the pioneering work is indeed exploratory.
I am a PhD student in Neuroscience working with humans. Some of my colleagues work with animals or cell cultures. One of my collaborators works in computational neuroscience and we are currently working on a new model together that we might be able to update based on the empirical findings in my lab. This type of collaboration is important, because the computational model "knows" more basic science, than I could possibly study in humans. That is why we can use computational models to test the limits of our findings - to test if our models of cognition are compatible with biophysical models, which are informed by principles derived from animal studies.